Lie symmetries of the Chow group of a Jacobian and the tautological subring

Let $J$ be the Jacobian of a smooth projective curve. We define a natural action of the Lie algebra of polynomial Hamiltonian vector fields on the plane, vanishing at the origin, on the Chow group $\operatorname{CH}(J)_{\mathbb{Q}}$. Using this action we obtain some relations between tautological cycles in $\operatorname{CH}(J)_{\mathbb{Q}}$.