Invariant star-products on symplectic manifolds

Let (M,F) be a symplectic manifold and consider a Lie subalgebra $\text{G}$ of its Lie algebra of symplectic vector fields. We prove that every one-differentiable deformation of order k of the Poisson Lie algebra of M, which is invariant with respect to $\text{G}$, extends to an invariant one-differentiable deformation of infinite order. If M admits a $\text{G}$-invariant linear connection, a similar result holds true for differentiable deformations and for star-products. In particular, if M admits a $\text{G}$- -invariant linear connection, there always exists a $\text{G}$-invariant star-product.