Symplectormophism groups of non-compact manifolds, orbifold balls, and a space of Lagrangians

We establish connections between contact isometry groups of certain contact manifolds and compactly supported symplectomorphism groups of their symplectizations. We apply these results to investigate the space of symplectic embeddings of balls with a single conical singularity at the origin. Using similar ideas, we also prove the longstanding expected result that the space of Lagrangian $\RR P^2$ in $T^*\RR P^2$ is weakly contractible.