I have the following system: $$ \begin{array}{ll} \dot{u} &= v\\ \dot{v} &= 2-u^2 - v^2 - u\end{array} $$ I need to determine an integral for this system, so a function $k(u,v)$ which is constant along orbits. Normally this is a Hamiltonian, but I heard that we can't define a Hamiltonian for this system. The hint is to define $w = v^2$ and solve for $\tfrac{dw}{du}$, but I don't see where I can get an equation containing $\tfrac{dw}{du}$.
I tried writing this as a 1-dimensional system, so $$ \ddot{u} - 2 + u^2 + (\dot{u})^2 + u = 0 $$ and multiplying this by $\dot{u}$, so we get $$ \ddot{u}\dot{u} - 2\dot{u} + u^2\dot{u} + (\dot{u})^2\dot{u} + u\dot{u} = 0, $$ I can integrate all the terms in this equation, except for the term $(\dot{u})^3$. Is there a nice way to resolve this problem?
Let $w = v^2$,
$$\begin{align} & \frac{dw}{du} = 2v\frac{dv}{du} = 2v\frac{\dot{v}}{\dot{u}} =2(2 - u^2 - w - u)\\ \iff & \left(\frac{d}{du} + 2 \right) w = 2(2 - u^2 - u)\\ \iff & \frac{d}{du}(e^{2u} w) = 2(2 - u^2 - u)e^{2u}\\ \iff & e^{2u} w = \int 2(2-u^2-u)e^{2u} du + \text{ const. } = (2-u^2) e^{2u} + \text{ const }\\ \implies& (u^2+v^2-2)e^{2u} = \text{ const. } \end{align} $$