Does a constant of motion always imply hamiltonian?

Question

If a dynamical system has a constant of motion, and is not already evidently Hamiltonian, is it always possible to use a change of variables and obtain a Hamiltonian system?

Edit: the constant of motion should be a function of all the variables in the system.

Answer 1

I doubt this is true in general. Take for example a system which consists of (uncoupled) oscillator and a dissipative system.

$\dot{x_1}=x_2$

$\dot x_2=-x_1$

$\dot{x_3}=-x_3$

$\dot{x_4}=-x_4$

This system preserves the energy of the oscillator (i.e. $x_1^2+x_2^2$ is a constant of motion), but is clearly non-Hamiltonian due to the dissipative nature of the second degree of freedom $(x_3,x_4)$. It is not even volume-preserving, which is a weaker property than being Hamiltonian.

Edit: Now if we want to have a constant of motion which is function of all states, then consider a transformation as follows:

$y_1 = x_1+x_3$

$y_2=x_2+x_4$

$y_3=x_3$

$y_4=x_4$

The transformed system becomes:

$\dot y_1=y_2-y_4-y_3$

$\dot y_2=-y_1+y_3-y_4$

$\dot y_3=-y_3$

$\dot y_4=-y_4$

This system will preserve $(y_1-y_3)^2+(y_2-y_4)^2$, but is not Hamiltonian.

I am not sure if this serves as a counter-example, but it may clarify some points.