What is an example of a connected symplectic manifold $(M,\omega)$, with a Hamiltonian action of $G = U(1) =S^{1}$ with infinitely many stabiliser types?


Infinitely many stabiliser types means that infinitely many sub-groups of $G$ appear as stabilisers as points in $M$.


I am aware that $M$ is necessarily non-compact.



Hamiltonian Group action with infinitely many stabiliser types
As an simple example with infinitely many different stabilizer subgroups you may take countably many disks. Since each disk may be rotated independently with a different speed, we obtain an action of $ U(1) $ such that all the subgroups $ \mathbb Z / n \mathbb Z$ with $ n \in \mathbb N $ occur as stabilizer.


Of course the example obtained in this way is disconnected and even has infinitely many connected components. However, one can use the same strategy to obtain a connected manifold with infinitely many distinct stabilizer groups. The idea is to embed the collection of disks in a bigger space and "smear" the action between them. Details can be found in an old paper ["On a problem of Montgomery" by C.T. Yang](http://www.jstor.org/stable/2033722).


