Let $T$ be a linear algebraic torus over $\mathbb C$ and $X$ be a smooth quasi-projective symplectic $T$-variety. Also, assume that the action of $T $ is free and $X/T$ exists as a smooth variety. Is this action Hamiltonian?

If the above action is Hamiltonian let us denote the algebraic moment map $X\rightarrow t^{\vee}$ by $\mu$. Does there exist a one-to-one correspondence, via $\mu$, between the symplectic leaves of $X/T$ and the coadjoint orbits of $T$ (which are just points of $t^{\vee}$ in this case)?