p-adic analogue of Eliasson–Vey normal form theorem

Establish whether a p-adic analogue of Eliasson–Vey’s normal form theorem holds for integrable Hamiltonian systems on p-adic analytic symplectic manifolds. Specifically, determine conditions under which there exist local symplectic coordinates and a local diffeomorphism so that the system is brought to a standard normal form near non-degenerate singularities (elliptic–elliptic, focus–focus, and rank-1 transversally elliptic), and formulate the precise p-adic statements and proofs.

Background

In the real analytic setting, Eliasson and Vey’s theorem provides a powerful local classification and linearization near non-degenerate singularities. The authors obtain linear normal forms in the p-adic case by direct computations but note the lack of a general theorem comparable to Eliasson–Vey.

A p-adic version would significantly advance the theory of p-adic integrable systems, providing structural results and simplifications similar to those known in the real case.