I would like some examples of ODEs (i.e., $\dot{x}=f(x)$, where $x\in\mathbb{R}^n$) that possess one or more rational first-integrals of the form $$H(x)=\frac{a_1^Tx+\alpha_1}{a_2^Tx+\alpha_2},$$ where $a_i\in\mathbb{R}^n$ and $\alpha_i\in\mathbb{R}$ are both constant. (Edit: It would also be very useful with examples of ODEs possessing higher-degree rational integrals, e.g., $H(x)=P(x)/Q(x)$ where $P(x)$ and $Q(x)$ are polynomial in $x$. )
Of course, you could always design an ODE whose flow preserves $H$, by writing $$\dot{x}=S\nabla H\in\mathbb{R}^n,$$ where $S=-S^T$ is a skew-symmetrix matrix. (In a similar way, you can also easily design an ODE that has multiple functions of the form $H$ as a first-integral). However, I would like an ODE that has been studied before or one with a physical application, e.g., where $H_i$ appears as a physical constraint, Hamiltonian function etc. Can someone point me to a reference for this, or a give a name of such a dynamical system?
Basically, I have stumbled across a method for solving such ODEs, and would like a nice example in my paper that is connected to the literature, instead of simply just an example that I've designed myself.