  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.
