I’m interested in finding interesting examples of non-Hamiltonian automorphisms of a symplectic manifold (not necessarily closed - but preferably so) $(M,\omega)$ that set-wise preserve a Lagrangian sub manifold $L$.
For example, given a Lagrangian sphere, we can define a (symplectic) Dehn twist along it which would set-wise preserve the sphere. Are there other examples like that?
Is such a property important for some reasons?