# Complex Germen on invariant isotropic tori under the Hamiltonian phases flow with in involution Hamilton functions

## TL;DR

This paper fully solves the problem of determining the existence and uniqueness of the complex germ on invariant isotropic tori under Hamiltonian flows with involutive functions, removing previous spectral restrictions.

## Contribution

It provides necessary and sufficient conditions for the complex germ's existence and uniqueness without requiring the simple spectrum condition.

## Key findings

- Established conditions for complex germ existence and uniqueness.
- Extended previous results to cases without simple spectrum.
- Analyzed Hamiltonian systems with cyclic variables.

## Abstract

M. M. Nekhoroshev put forward the problem of to find the Complex Germ on a isotropic invariant torus with respect to Hamiltonian phases flows which come from k-functions in involution. This statement was partially solved in [9] establishing that if certain simplectic operator has a simple spectrum then the complex germ exist. In this work we solve this problem, providing a full solution, i.e. we present conditions for the existence and uniqueness of complex germ through the monodromy operator constructed in [9], but without the simple spectrum condition. We study also the Hamiltonian system with cyclic variables.