Let $(M,\omega)$ be a symplectic manifold with a hamiltonian effective torus action. Suppose it has an isolated fixed point $p$. Is it true that there exists an invariant neighborhood $U$ of $p$ such that the action on it is linear e.g. $U$ is symplectomorphic to an open ball in $\mathbb C^n$ with standard symplectic form and the torus action is linear? What can be said about local action around fixed submanifolds of dimension greater than $0$?
P.Pushkar told me that it must be true and I can read about it in the journal "Functional analysis and its applications" (Функциональный анализ и его применения) but unfortunately he didn't seem to manage to tell me the author, the name and the year of the publication. Perhaps you can find it.
There is always an equivariant local symplectomorphism with $T_pM$ with its 2-form and linear isotropy action, by the Moser-Weinstein proof. But that constant 2-form then has more possible “equivariant normal forms” than just $\sum dp_i\wedge dq_i$ — see e.g. Dellnitz-Melbourne (1993; pdf).