Geometry and solutions of an epidemic SIS model permitting fluctuations and quantization

Let $N$ be a manifold. A time-dependent vector field on $N$ is a differentiable mapping

(2.1)

$$X:\mathbb{R}\times N\longrightarrow TN,\qquad\tau_{N}\circ X=pr_{2}$$

where $pr_{2}$ is the projection to the second factor in $\mathbb{R}\times N$. In this regard, we can consider a time-dependent vector as a set (family) of standard vector fields $\{X_{t}\}_{t\in\mathbb{R}}$, depending on a single parameter. The converse of this assertion is also true, that is a set of vector fields depending smoothly on a single parameter can be written as a time-dependent vector field [18]. We plot the following diagram to summarize the discussions.

(2.2)

A time dependent vector field determines a non-autonomous ODE system

(2.3)

$$\frac{dx}{dt}=X(t,x),\qquad x\in N.$$

The suspension of a time-dependent vector field $X$ on $N$ is the vector field on $\mathbb{R}\times N$ defined to be $X+\partial/\partial t$, where $t$ is the global coordinate for $\mathbb{R}$. An integral curve of a time-dependent vector field $X$ is an integral curve of the suspension [3]. Conversely, every system of first-order differential equations in normal form describes the integral curves of a unique time-dependent vector field.

Consider the case that $N$ is a two dimensional manifold with local coordinates $(q,p)$, and the extended space $\mathbb{R}\times N$ with induced coordinates $(t,q,p)$. A time dependent vector field is written as

(2.4)

$$X=k(t,q,p)\frac{\partial}{\partial q}+g(t,q,p)\frac{\partial}{\partial q}$$

where $k$ and $g$ are real valued functions on $\mathbb{R}\times N$. In this coordinate system, the dynamical equations governed by $X$ determine a system of nonautonomous ordinary differential equations

(2.5)

$$\dot{q}=k(t,q,p),\qquad\dot{p}=g(t,q,p).$$

Notice that, the SISf model (1.5) fits this picture with even the generalization $\rho_{0}=\rho(t)$. We shall turn to this discussion in the following subsections.

Vessiot-Guldberg algebra. Start with a time-dependent vector field $X$, and determine the set $\{X_{t}\}_{t\in\mathbb{R}}$ of vector fields. We denote the smallest Lie subalgebra containing the set $\{X_{t}\}_{t\in\mathbb{R}}$ by $V^{X}$. The associated distribution $\mathcal{D}^{X}$ for a time-dependent vector field $X$ is the generalized distribution spanned by the vector fields in $V^{X}$ that is

(2.6)

$$\mathcal{D}^{X}_{x}=\{Y_{x}\mid Y\in V^{X}\}\subset T_{x}N.$$

The associated codistribution $(\mathcal{D}^{X})^{\circ}$ for a time-dependent vector field $X$ is then defined to be the annihilator of $\mathcal{D}^{X}$ that is

(2.7)

$$(\mathcal{D}^{X}_{x})^{\circ}=\{\vartheta\in T_{x}^{*}N\mid\vartheta(Z_{x})=0,\quad\forall Z_{x}\in\mathcal{D}_{x}^{X}\}.$$

is a subbundle of the cotangent bundle $T^{*}N$. Our interest relies in a distribution $\mathcal{D}^{X}$ determined by a finite-dimensional $V^{X}$ and hence $\mathcal{D}^{X}$ becomes integrable on the whole $N$. In this case, $V^{X}$ is called Vessiot-Guldberg Lie algebra. It is worth noting that even in this case, $(\mathcal{D}^{X})^{\circ}$ does not need to be a differentiable codistribution.

A function $f$ on an open neighbourhood $U$ is a local $t$-independent constant of motion for a system $X$ if and only if $df(x)$ in $(\mathcal{D}^{X}_{x})^{\circ}|_{U}$ for all $x$ in $N$. This statement reads that (locally defined) time-independent constants of motion of time-dependent vector fields are determined by (locally defined) exact one-forms taking values in the associated codistribution. Therefore, $(\mathcal{D}^{X})^{\circ}$ is a crucial object in the calculation of such constants of motion for a system $X$.

Let $X$ be a time-dependent vector field defined on a manifold $N$. A superposition rule depending on $m$ particular solutions of $X$ is a function

(2.8)

$$\phi:N^{m}\times N\rightarrow N,\qquad x=\phi(x_{(1)},\dots,x_{(m)},\lambda)$$

such that the general solution $x(t)$ of $X$ can be written as $x(t)=\phi(x_{(1)}(t),\dots,x_{(m)}(t),\lambda)$ for any generic family $x_{(1)}(t),\dots,x_{(m)}(t)$ of particular solutions. Here, $\lambda$ is a point of $N$ related with the initial conditions. A system of equations admitting superposition rule is called Lie system.

Lie–Scheffers Theorem. A modern statement of this result is described in [16]. A first-order system (2.3) admits a superposition rule if and only if the associated time-dependent vector field $X$ can be written as

(2.9)

$$X_{t}={{\sum_{\alpha=1}^{r}}}b_{\alpha}(t)X_{\alpha}$$

for a certain family $b_{1}(t),\dots,b_{r}(t)$ of time-dependent functions and a family $X_{1},\dots,X_{r}$ of vector fields on N spanning an $r$-dimensional real Lie subalgebra of vector fields. We refer to [35] for details, see [58, 50] for some further discussions and the first examples. The Lie–Scheffers Theorem yields that a system $X$ admits a superposition rule if and only if $V^{X}$ is finite-dimensional, [18].

An itinerary map to derive superposition rules. General solutions of Lie systems can also be investigated through superposition rules. There exist various procedures to derive them [4, 57], but we hereafter use the method devised in [16], which is based on the notion of diagonal prolongation [18]. Let us denote the $m+1$-times Cartesian product of $N$ by itself as $N^{(m+1)}$. We denote by

(2.10)

$${\rm pr}:N^{(m+1)}\longrightarrow N,\qquad(x_{(0)},x_{(1)},\ldots,x_{(m)})\mapsto x_{(0)}$$

the projection onto the first factor. Given a time-dependent vector field $X$ on $N$, the diagonal prolongation $\widetilde{X}$ of $X$ to the product space $N^{(m+1)}$ is a unique time-dependent vector field on $N^{(m+1)}$ projecting to $X$ by the projection ${\rm pr}$ that is ${\rm pr}_{*}\widetilde{X}_{t}=X_{t}$ for all $t$. We also require $\widetilde{X}$ to be invariant under the permutations $x_{(i)}\leftrightarrow x_{(j)}$ with $i,j=0,\dots,m$. The procedure to determine superposition rules described in [16] goes as follows:

Step 1. Take a basis $X_{1},\dots,X_{r}$ of a Vessiot–Guldberg Lie algebra associated with the Lie system $X$.

Step 2. Choose the minimum integer $m$ so that the diagonal prolongations $\widetilde{X}_{1},\ldots,\widetilde{X}_{r},$ of $X_{1},\dots,X_{r}$ to $N^{m}$ are linearly independent at a generic point.

Step 3. Obtain, for instance, by the method of characteristics, $n$ functionally independent first-integrals $F_{1},\ldots,F_{n}$ common to all the diagonal prolongations, $\widetilde{X}_{1},\ldots,\widetilde{X}_{r},$ to $N^{(m+1)}$. We require such functions to satisfy

$$\frac{\partial(F_{1},\dots,F_{n})}{\partial\left((x_{1})_{(0)},\dots,(x_{n})_{(0)}\right)}\neq 0.$$

Assume that these integrals take certain constant values, i.e., $F_{i}=k_{i}$ where $i=1,\ldots,n$, and employ these equalities to express the variables $(x_{1})_{(0)},\dots,(x_{n})_{(0)}$ in terms of the variables of the other copies of $N$ within $N^{(m+1)}$ and the constants $k_{1},\ldots,k_{n}$.

The obtained expressions constitute a superposition rule in terms of any generic family of $m$ particular solutions and $n$ constants. Let us apply these steps in the realm of the SISf model.

The model (1.5) can be generalized to a model represented by a time-dependent vector field

(2.11)

$$X_{t}=\rho_{0}(t)X_{1}+X_{2}$$

where the constitutive vector fields are computed to be

(2.12)

$$X_{1}=q\frac{\partial}{\partial q}-p\frac{\partial}{\partial p},\quad X_{2}=\left(-q^{2}-\frac{1}{p^{2}}\right)\frac{\partial}{\partial q}+2qp\frac{\partial}{\partial p}.$$

The generalization comes from the fact that $\rho_{0}(t)$ is no longer a constant, but it can evolve in time. Let us apply the steps introduced in the previous section one by one to arrive at the general solution.

Step 1. For the vector fields in (2.12), a direct calculation shows that the Lie bracket

(2.13)

$$[X_{1},X_{2}]=X_{2}$$

is closed within the Lie algebra. This implies that the SISf model (1.5) is a Lie system. The Vessiot-Guldberg algebra spanned by $X_{1},X_{2}$ is an imprimitive Lie algebra of type $I_{14}$ according to the classification presented in [8].

Step 2. If we copy the configuration space twice, we will have four degrees of freedom $(q_{1},p_{1},q_{2},p_{2})$ and we will archieve precisely two first-integrals in vinicity of the Fröbenius theorem. A first-integral for $X_{t}$ has to be a first-integral for $X_{1}$ and $X_{2}$ simultaneously. We define the diagonal prolongation $\widetilde{X}_{1}$ of the vector field $X_{1}$ in the decomposition (2.13). Then we look for a first integral $F_{1}$ such that $\widetilde{X}_{1}[F_{1}]$ vanishes identically. Notice that if $F_{1}$ is a first-integral of the vector field $\widetilde{X}_{1}$ then it is a first integral of $\widetilde{X}_{2}$ due to the commutation relation. For this reason, we start by integrating the prolonged vector field

(2.14)

$$\widetilde{X}_{1}=q_{1}\frac{\partial}{\partial q_{1}}+q_{2}\frac{\partial}{\partial q_{2}}-p_{1}\frac{\partial}{\partial p_{1}}-p_{2}\frac{\partial}{\partial p_{2}}$$

through the following characteristic system

(2.15)

$$\frac{dq_{1}}{q_{1}}=\frac{dq_{2}}{q_{2}}=\frac{dp_{1}}{-p_{1}}=\frac{dp_{2}}{-p_{2}}.$$

Fix the dependent variable $q_{1}$ and obtain a new set of dependent variables, say $(K_{1},K_{2},K_{3})$, which are computed to be

(2.16)

$$K_{1}=\frac{q_{1}}{q_{2}},\qquad K_{2}=q_{1}p_{1},\qquad K_{3}=q_{1}p_{2}.$$

Step 3. This induces the following basis in the tangent space

(2.17)

$$\begin{split}\frac{\partial}{\partial K_{1}}=q_{2}\frac{\partial}{\partial q_{1}}-\frac{q_{2}p_{1}}{q_{1}}\frac{\partial}{\partial p_{1}}-\frac{q_{2}p_{2}}{q_{1}}\frac{\partial}{\partial p_{1}},\qquad\frac{\partial}{\partial K_{2}}=\frac{1}{q_{1}}\frac{\partial}{\partial p_{1}},\qquad\frac{\partial}{\partial K_{3}}=\frac{1}{q_{1}}\frac{\partial}{\partial p_{2}}.\end{split}$$

provided that $q_{1}$ is not zero. Introducing the coordinate changes exhibited in (2.16) into the diagonal projection $\widetilde{X}_{2}$ of the vector field $X_{2}$, we arrive at the following expression

$$\begin{split}\widetilde{X}_{2}=&\left(2K_{1}-\left(1+\frac{1}{K_{1}^{2}}\right)\right)\frac{\partial}{\partial K_{1}}+\left(\left(\frac{1}{K_{2}^{2}}+\frac{1}{K_{3}^{2}}\right)K_{2}^{2}-\left(1+\frac{1}{K_{1}^{2}}\right)K_{2}\right)\frac{\partial}{\partial K_{2}}\\ &+\left(2\frac{K_{3}}{K_{2}}-\left(1+\frac{1}{K_{1}^{2}}\right)K_{3}\right)\frac{\partial}{\partial K_{3}}.\end{split}$$

To integrate the system once more, we use the method of characteristics again and obtain

(2.18)

$$\frac{d\ln{|K_{1}|}}{1-\frac{1}{K_{1}^{2}}}=\frac{d\ln{|K_{2}|}}{\frac{1}{K_{2}}+\frac{K_{2}}{K_{3}}-1-\frac{1}{K_{1}^{2}}}=\frac{d\ln{|K_{3}|}}{\frac{2}{K_{2}}-\left(1+\frac{1}{K_{1}^{2}}\right)}.$$

Exact solution. We obtain two first integrals by integrating in pairs $(K_{1},K_{2})$ and $(K_{1},K_{3})$, where we have fixed $K_{1}$. After some cumbersome calculations we obtain

(2.19)

$$K_{2}=\frac{K_{1}\left(4k_{2}^{2}K_{1}^{2}+4k_{1}k_{2}K_{1}+k_{1}^{2}-4\right)}{2(K_{1}+1)(K_{1}-1)k_{2}(2k_{2}K_{1}+k_{1})},\qquad K_{3}=\frac{K_{1}\left(k_{2}K_{1}^{2}+k_{1}K_{1}+\frac{k_{1}^{2}-4}{4k_{2}}\right)}{(K_{1}+1)(K_{1}-1)}.$$

By substituting back the coordinate transformation (2.16) into the solution (2.19) (please notice the difference between capitalized constants $(K_{1},K_{2},K_{3})$ and lower case constants $(k_{1},k_{2})$, we arrive at the following implicit equations

(2.20)

$$\begin{split}q_{1}&=-\frac{q_{2}\left(k_{1}k_{2}\pm\sqrt{4k_{2}^{2}p_{2}^{2}q_{2}^{2}+k_{1}^{2}k_{2}p_{2}q_{2}-4k_{2}^{3}p_{2}q_{2}-4k_{2}p_{2}q_{2}+4k_{2}^{2}}\right)}{2k_{2}(-p_{2}q_{2}+k_{2})}\\ p_{1}&=\frac{4q_{1}^{2}k_{2}^{2}+4q_{1}q_{2}k_{1}k_{2}+q_{2}^{2}k_{1}^{2}-4q_{2}^{2}}{2k_{2}(2q_{1}^{3}k_{2}+q_{1}^{2}k_{1}q_{2}-2q_{1}k_{2}q_{2}^{2}-k_{1}q_{2}^{3})}.\end{split}$$

Let us notice that the equations (2.20) depend on a particular solution $(q_{2},p_{2})$ and two constants of integration $(k_{1},k_{2})$ which are related to initial conditions.

Let us show now the graphs and values of the initial conditions for which the solution reminds us of sigmoid behavior, which is the expected growth of $\rho(t)$. As particular solution for $(q_{2},p_{2})$, we have made use of particular solution 2 given in Figure 2 through its corresponding values of $q,p$ through the change of variables $q=\langle\rho\rangle$ and $p=1/\sigma$.

Notice that we have not included the error graph since it gives a constant zero graph because $k_{1}\rightarrow 0$

Step 3 revisited - linear approximation. Since the solution (2.20) is quite complicated, one may look for a solution of a linearized model. We first employ the following change of coordinates

(2.21)

$$\{u=\ln{|K_{1}|},v=\ln{|K_{2}|},w=\ln{|K_{3}|}\}.$$

In terms of these new variables, the system (2.18) reads

(2.22)

$$\frac{du}{1-e^{-2u}}=\frac{dv}{e^{-v}+e^{v-w}-1-e^{-2u}}=\frac{dw}{2e^{-v}-(1+e^{-2u})}.$$

One can solve the system above by introducing a linear approximation

(2.23)

$$\begin{split}1-e^{-2u}&\simeq 2u,\\ e^{-v}+e^{v-w}-1-e^{-2u}&\simeq 2u-w,\\ 2e^{-v}-(1+e^{-2u})&\simeq 2u-2v,\end{split}$$

after which (2.22) reads

(2.24)

$$\frac{du}{2u}=\frac{dv}{2u-w}=\frac{dw}{2u-2v}.$$

We can solve now $v$ and $w$ in terms of $u$ and obtain

(2.25)

$$v(u)=k_{1}u^{-\sqrt{2}/2}+k_{2}u^{\sqrt{2}/2}+u,\qquad w(u)=\sqrt{2}\left(k_{1}u^{-\sqrt{2}/2}-k_{2}u^{\sqrt{2}/2}\right).$$

According to the algorithm for deriving superposition rule, we need to isolate the constants of integration $k_{1}$ and $k_{2}$. Hence, the two first integrals read now

(2.26)

$$k_{1}=u^{\frac{\sqrt{2}}{2}}\big{(}\sqrt{2}v-\sqrt{2}u+w\big{)}/{2\sqrt{2}},\qquad k_{2}=u^{-\frac{\sqrt{2}}{2}}\big{(}\sqrt{2}v-\sqrt{2}u-w\big{)}/2\sqrt{2}.$$

Now, if we substitute the coordinate changes in (2.21) and in (2.16), we arrive at the following general solution

(2.27)

$$q_{1}=q_{2}\exp\Big{(}-\frac{\ln{(q_{2}p_{2})}}{1+k_{1}+k_{2}}\Big{)},\qquad p_{1}=\frac{1}{q_{2}}\exp\Big{(}\frac{\sqrt{2}}{2}\frac{(k_{1}-k_{2})\ln{(q_{2}p_{2})}}{1+k_{1}+k_{2}}\Big{)}.$$

which can be written as

(2.28)

$$q_{1}=q_{2}\Big{(}q_{2}p_{2}\Big{)}^{\frac{-1}{1+k_{1}+k_{2}}},\qquad p_{1}=\frac{1}{q_{2}}\Big{(}q_{2}p_{2}\Big{)}^{\frac{\sqrt{2}}{2}\frac{k_{1}-k_{2}}{1+k_{1}+k_{2}}}.$$

Notice that the solution depends on a particular solution $(q_{2},p_{2})$ and two constants of integration $(k_{1},k_{2})$, as in (2.20).

Let us show now the graphs and values of the initial conditions for which the solution reminds us of sigmoid behavior, which is the expected growth of $\rho(t)$. As particular solution for $(q_{2},p_{2})$, we have made use of particular solution 2 given in Figure 2 through its corresponding values of $q,p$ through the change of variables $q=<\rho>$ and $p=1/\sigma$.

A Lie-Hamiltonian structure is a triple $(N,\Lambda,h)$, where $(N,\Lambda)$ stands for a Poisson manifold and $h$ represents a $t$-parametrized family of functions $h_{t}:N\rightarrow\mathbb{R}$ such that the Lie algebra $\mathcal{H}:={\rm Lie}(\{h_{t}\}_{t\in\mathbb{R}})$ generated by this family is finite-dimensional Poisson algebra. A time-dependent vector field $X$ is said to possess a Lie-Hamiltonian structure $(N,\Lambda,h)$ if $X_{t}$ is the Hamiltonian vector field corresponding to $h_{t}$, for each $t\in\mathbb{R}$,

(3.1)

$$X_{t}=-\hat{\Lambda}\circ d(h_{t}).$$

In this case, $X$ is called a Lie-Hamilton system, and $\mathcal{H}$ is called a Lie–Hamilton algebra for $X$. Now we examine superposition rules for the case of Lie–Hamilton systems.

Itinerary map for superposition rules of the Lie–Hamilton Systems. Let $X$ be a Lie–Hamilton system with a Poisson algebra $\mathcal{H}$ spanned by the linearly independent Hamiltonian functions $\{h_{1},\dots,h_{r}\}$.

Step 1. Consider the injection $\mathfrak{g}\hookrightarrow{\mathcal{H}}$ of a Lie algebra $\mathfrak{g}$ into $\mathcal{H}$ that turns the basis $v_{i}$ of $\mathfrak{g}$ into the Hamiltonian functions $\phi(v_{i}):=h_{i}$ for $i=1,\ldots,r$. Referring to this inclusion, we can define Poisson algebra homomorphisms

(3.2)

$$D:S(\mathfrak{g})\to C^{\infty}(N),\qquad D^{(m)}:S^{(m)}(\mathfrak{g})\to C^{\infty}(N)^{(m)}\subset C^{\infty}(N^{m}),$$

where $S(\mathfrak{g})$ is the symmetric algebra.

Step 2. If $\mathcal{C}$ is a polynomial Casimir of the Poisson algebra $S(\mathfrak{g})$, say $\mathcal{C}=\mathcal{C}(v_{1},\dots,v_{r})$, then $D(\mathcal{C})$ is a constant of motion for $X$, and so are the functions defined by

(3.3)

$$F^{(k)}(h_{1},\dots,h_{r})=D^{(k)}\left[\Delta^{(k)}\left({C(v_{1},\dots,v_{r})}\right)\right],\qquad k=2,\ldots,m.$$

Let us notice, that each $F^{(k)}$ can naturally be considered as a function of $C^{\infty}(M^{m})$ for every $m\geq k$.

Step 3. From the functions $F^{(k)}$, we can obtain other constants of motion in the form

(3.4)

$$F_{ij}^{(k)}=S_{ij}(F^{(k)}),\qquad 1\leq i<j\leq k,\qquad k=2,\ldots,m,$$

where $S_{ij}$ is the permutation of variables $x_{(i)}\leftrightarrow x_{(j)}$ on $M^{m}$.

Indeed, since the prolongation $\widetilde{X}$ is invariant under the permutations $x_{(i)}\leftrightarrow x_{(j)}$, then the $F_{ij}^{(k)}$ are also $t$-independent constants of motion for the diagonal prolongations $\widetilde{X}$ to $M^{m}$. Let us now illustrate this procedure in the following example.

The SISf epidemic system in (1.5) admits a Hamiltonian formulation as in (1.9), where the corresponding Hamilton equations are (1.5). In this section we will retrieve the Hamiltonian (1.9) using the theory of Lie systems, in particular, the theory of Lie–Hamilton systems. To retrieve the Hamiltonian, first we need to prove that equations in (1.5) form a Lie–Hamilton system.

We have already proven in (2.13) that (1.5) defines a Lie system. In order to see if it is a Lie–Hamilton system, we first need to check whether the vector fields in (2.13) are Hamiltonian vector fields. Consider now the canonical symplectic form $\omega=dq\wedge dp$. It is easy to check that the vector fields $X_{1}$ and $X_{2}$ in (2.13) are Hamiltonian with respect to the Hamiltonian functions

(3.10)

$$h_{1}=-qp,\quad\quad h_{2}=-q^{2}p+\frac{1}{p},$$

respectively. It is easy to see that the Poisson bracket of these two functions reads $\{h_{1},h_{2}\}=h_{2}$. It means that the Hamiltonian functions form a finite dimensional Lie algebra, denoted in the literature as $I^{r=1}_{14A}\simeq\mathbb{R}\ltimes\mathbb{R}$, and it is isomorphic to the one defined by vector fields $X_{1},X_{2}$. The Hamiltonian function for the total system is

(3.11)

$$h=\rho_{0}(t)h_{1}+h_{2}=-q^{2}p+\frac{1}{p}-\rho_{0}(t)qp$$

and it is exactly the Hamiltonian function (1.9) proposed in [43].

Lie-Hamilton systems can also be integrated in terms of a superposition rule, as it was explained in the preliminary section. In order to do that, we need to find a Casimir function for the Poisson algebra, but unfortunately, there exists no nontrivial Casimir in this particular case. It is interesting to see how a symmetry of the Lie algebra $\{X_{1},X_{2}\}$ commutes with the Lie bracket, i.e. the vector field

(3.12)

$$Z=-\frac{1}{2}\frac{p(C_{2}p^{2}q^{2}+4C_{1}pq+C_{2})}{(pq-1)(pq+1)}\frac{\partial}{\partial p}+\frac{C_{1}p^{4}q^{4}+C_{2}p^{3}q^{3}-C_{2}pq-C_{1}}{p(pq-1)^{2}(pq+1)^{2}}\frac{\partial}{\partial q}$$

fulfills $[X_{1},Z]=0,\quad[X_{2},Z]=0.$ Notice too that $Z$ is a conformal vector field, that is,

(3.13)

$$\mathcal{L}_{Z}\omega=-(C_{2}/2)\omega.$$

Since it is a Hamiltonian system, one would expect that a first integral for $Z$, let us say $f$, would Poisson commute with the Poisson algebra $\{h_{1},h_{2}\}$, since $Z=-\hat{\Lambda}(df)$. Nonetheless, this is not the case unless $f=\text{constant}$. This implies that the Casimir is a constant, hence trivial and the coalgebra method can not be directly applied. However, there is a way in which we can circumvent this problem by considering an inclusion of the algebra $I^{r=1}_{14A}$ as a Lie subalgebra of a Lie algebra to another class admitting a Lie–Hamiltonian algebra with a non-trivial Casimir. In this case, we will consider the algebra, denoted by $I_{8}\simeq\mathfrak{iso}(1,1)$, due to the simple form of its Casimir. If we obtain the superposition rule for $I_{8}$, we simultaneously obtain the superposition for $I_{14A}^{r=1}$ as a byproduct.

The Lie–Hamilton algebra $\mathfrak{iso}(1,1)$ has the commutation relations

(3.14)

$$\{h_{1},h_{2}\}=h_{0},\quad\{h_{1},h_{3}\}=-h_{1},\quad\{h_{2},h_{3}\}=h_{2},\quad\{h_{0},\cdot\}=0,$$

with respect to $\omega=dx\wedge dy$ in the basis $\{h_{1}=y,h_{2}=-x,h_{3}=xy,h_{0}=1\}$. The Casimir associated to this Lie–Hamilton algebra is $\mathcal{C}=h_{1}h_{2}+h_{3}h_{0}$. Let us apply the coalgebra method to this case. Mapping the representation without coproduct, the first iteration is trivial, i.e., $F=0$. We could usethe second order coproduct and third order coproduct $\Delta^{(2)}$ and $\Delta^{(3)}$, or the second order coproduct $\Delta^{(2)}$ together with the permuting subindices property. We need three constants of motion, this would be equivalent to integrating the diagonal prolongation $\widetilde{X}$ on $(\mathbb{R}^{2})^{3}$. Using the coalgebra method and subindex permutation, one obtains

(3.15)

$$\begin{split}F^{(2)}&=(x_{1}-x_{2})(y_{1}-y_{2})=k_{1},\\ F_{23}^{(2)}&=(x_{1}-x_{3})(y_{1}-y_{3})=k_{2},\\ F_{13}^{(2)}&=(x_{3}-x_{2})(y_{3}-y_{2})=k_{3}.\end{split}$$

From them, we can choose two functionally independent constants of motion. Our choice is $F^{(2)}=k_{1},F^{(2)}_{23}=k_{2}$. The introduction of $k_{3}$ simplifies the final result, with expression

(3.16)

$$\begin{split}x_{1}(x_{2},y_{2},x_{3},y_{3},k_{1},k_{2},k_{3})&=\frac{1}{2}(x_{2}+x_{3})+\frac{k_{2}-k_{1}\pm B}{2(y_{2}-y_{3})},\\ y_{1}(x_{2},y_{2},x_{3},y_{3},k_{1},k_{2},k_{3})&=\frac{1}{2}(y_{2}+y_{3})+\frac{k_{2}-k_{1}\mp B}{2(x_{2}-x_{3})},\end{split}$$

where

(3.17)

$$B=\sqrt{k_{1}^{2}+k_{2}^{2}+k_{3}^{2}-2(k_{1}k_{2}+k_{1}k_{3}+k_{2}k_{3})}.$$