Poisson ideals and the Witt algebra

The goal of my talk is to provide a description of Poisson ideals of the universal enveloping algebra of the Witt algebra (this is a ‘half’ of the Lie algebra of vector fields on a one-dimensional torus) and of several similar algebras.
In my talk I will show that, for a two-sided ideal $I$, the quotient by $I$ has a finite Gelfand–Kirillov dimension and the variety $\operatorname{Var}(I)$ attached to $I$ can be identified with a subset of the set of (one-sided) recursive sequences.
This poses an interesting problem on ‘orbits’ of the coadjoint representation of the Lie algebra of vector fields on a torus, where the acting group coincides with the (local) group of diffeomorphisms of a torus generated by the integrals of vector fields on it (I wish to mention that this group has no definition in a purely algebraic setting).