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

Amaury Alvarez Cruz, Baldomero Valiño Alonso

**Keywords:**

In this work we give the answer to the question about the existence and uniqueness of a complex germ on a isotropic torus invariant

respect to the Hamiltonian flows defined by k function $F_1,\ldots,F_k$ that stay in involution in the phase state consisting a manifold $M^{2n}$.

We proof that there exist such germ if and only if the reduce monodromy operator with period

$T_j$, $j=1,\dots,k$ are stable. This germ is unique if at least one operator is strong stable.

The result obtained here were applied to the case of an Hamiltonian with cyclic variables resulting in new condition for the

existence and uniqueness of complex germ.