Inspired by the results on symmetries of the symplectic Dirac operator, we realize symplectic spinor fields and the symplectic Dirac operator in the framework of (the double cover of) homogeneous projective structure in two real dimensions. The symmetry group of the homogeneous model of the double cover of projective geometry in two real dimensions is $\tilde{}(3,)$.

In this paper, $M$ denotes a smooth manifold of dimension $n$, $A$ a Weil algebra and $M^A$ the associated Weil bundle. When $(M,{\omega }_M)$ is a Poisson manifold with $2$-form ${\omega }_M$, we construct the $2$-Poisson form ${\omega }_{M^A}^A$, prolongation on $M^A$ of the $2$-Poisson form ${\omega }_M$. We give a necessary and sufficient condition for that $M^A$ be an $A$-Poisson manifold.