Let $(M, \omega)$ be a symplectic manifold, $H$ a Hamiltonian function on $M$, $Y = H^{-1}(c)$ for a regular value $c$, and $J$ a compatible almost complex structure.

If $X_H$ is the Hamiltonian vector field associated to $H$, then $-JX_H$ is the gradient of $H$ with respect to the metric associated to $\omega$ and $J$.

Under what conditions will $Y$ be of contact type with a (local) Liouville vector field given near $Y$ by $-JX_H$?

Under what conditions will there exist any contact structure on $Y$ such that $X_H$ is the Reeb vector field?

Note: I'm fine with answers "up to scaling" - i.e. multiplying the metric, the symplectic form, or any of the vector fields by a nonvanishing smooth function.