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What is an example of a 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.

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.

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

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

What is an example of a 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.