Consider the two-dimensional autonomous system \begin{equation*}\vec{\frac{dx}{dt}}(t)=\vec{f}(\vec{x}(t)):\mathbb{R}^{2}\to\mathbb{R}^{2}.\end{equation*}
Suppose the system has a conserved quantity of motion associated with it (i.e. an energy conservation law): \begin{equation*}E(\vec{x}(t))=E_{0}\;\forall\;t\Longrightarrow\frac{dE}{dt}=0.\end{equation*}
Then here are some questions.
(1) Is it possible to have a second conserved quantity of motion which is different from $E$?
(2) Is it necessarily the case that $E(x_{1},x_{2})=E_{0}$ corresponds to the integral family of curves which satisfes \begin{equation*}\frac{dx_{2}}{dx_{1}}=\frac{f_{2}(x_{1},x_{2})}{f_{1}(x_{1},x_{2})}\mapsto F(x_{1},x_{2})=E_{0}\end{equation*} (i.e. the solution curves in the phase-plane?). For conservative mechanical systems such as $x''=-f(x)$ this is evidently the case (if one uses the standard physical conservation of energy). But what about for general systems. And why or why not is this the case (i.e. a proof)?
Actually, I think I can prove (2): If $E$ represents the conserved quantity, then along trajectories, $E(\vec{x})=E_{0}\;\forall\;t$, so that the trajectories must correspond to the level sets of the function $E(\vec{x}):\mathbb{R}^{2}\to\mathbb{R}$. From another point of view, the initial conditions determine (and are determined by) the initial energy $E_{0}$. Then...
(3) Assuming (2) is indeed true, then if the system has a conservative quantity, presumably it can always be obtained by integrating the differential equation obtained by eliminating $t$ from the system? Of course there are other ways to obtain energy functions (for example, in the mechanical system one can multiply both sides of the equation by $\frac{dx}{dt}$ and integrate). But what (2) tells me, is that any method which is used to obtain $E$ is necessarily equivalent to integrating the trajectories.
(4) Assuming (3), once one integrates the t-eliminated ODE, then all one must do is check that $\frac{dE}{dt}=0$. If this holds, then the system is conservative and you have your energy function (I am tacitly assuming (1) is NOT true; i.e. that the conserved quantity is unique). If it does not, then you have your trajectories yes, but there is no conserved quantity along such trajectories.
I might add some more questions to this later, but I'll leave it at that for now.
Thanks!
EDIT:
(4) Was kind of a dumb question/observation. Of course, when you get the integral family of curves you get something like $F(x_{1},x_{2})=C$. Differentiating this with respect to $t$ of course gives you $0$, so $F$ is obviously constant along trajectories. But doesn't this effectively say every system is conservative then? ... Now I'm confused. ><