# Gradient system but not Hamiltonian

We know that Hamiltonian is gradient system if and only if $H$ is harmonic. So we can easily find an example that is Hamiltonian but not gradient. But this proposition does not say every gradient system is Hamiltonian. Is there any examples?

It's pretty easy to cook up such example. Take $V(x, y) = -\frac{1}{2}(x^2+y^2)$ and consider the system $\dot{x} = \frac{\partial V}{\partial x} = -x, \; \dot{y} = \frac{\partial V}{\partial y} = -y$. This system is gradient, but it can't be a Hamiltonian or even a conservative system: having asymptotically Lyapunov stable equilibrium (namely, the one at the origin) forbids having non-trivial first integral.