I'm currently working in the endemic SIRS bihamiltonian structure such that the Poisson brackets, Hamiltonian and Casimir are given by: $$H=S+I+R$$ $$\left\{ S, I\right\}= 0 \hspace{2cm} \left\{S,R\right\}=-\beta SI+\mu I \hspace{2cm} \left\{I,R\right\}=\beta SI - (\alpha+\mu)I$$ $$\mathcal{C} = S+I-\frac{\alpha}{\beta} \ln(\beta S-\mu).$$
I was able to linealize the structure obtaining: $$H'=X_1-\alpha X_3+ \frac{\alpha}{\beta}\frac{X_3}{X_1}(e^{\frac{\beta}{\alpha}}+\mu)$$ $$\left\{ X_1, X_2\right\}= 0 \hspace{2cm} \left\{X_2,X_3\right\}=X_1 \hspace{2cm} \left\{X_1,X_3\right\}=X_1$$ $$\mathcal{C}' = X_1-X_2.$$
But now I'm having trouble while trying to compare this structures of the Listed Real Lie algebras in Patera's article.
So, I was wondering if I should reduce the structure of the algebra I have, or what should be my next step. Any insights would be really appreciated.
