arXiv:1905.13169v1 [math.SG] 13 May 2019 Complex Germen on invariant isotropic tori under the Hamiltonian phases flow with in involution Hamilton functions A. C. Alvarez * , Baldomero Vali˜ no Alonso † May 31, 2019 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. 1 Introduction In several problems of the quantum and theoretical physics approximated solu- tion to the partial differential equations which contain a small parameter in the higher derivative order are obtained, as well as to approximated eigenvalues and eigenvector of self-adjoint differential operator which depend on a small param- eter. In such problems have been used the asymptotic methods [8, 7, 6], which nowadays are developed widely in several branches of the physics-mathematics. It is well known the success of asymptotic methods, e.g. with the quantifica- tion method was solved the older and sharp problem of the mechanic classic: calculation of the energetic level of the hydrogen atom [3]. In [7] over a 2n-dimensional phases space to obtain an asymptotic quasiclassi- cal solution with respect to a small parameter on an isotropic tori k-dimensional (k<n) is obtained. This asymptotic on a torus is accomplished with a new geometric object which was called the Complex Germen .i.e. a family of com- plex planes with certain properties. Such object does not exist over any isotropic * Instituto Nacional de Matem´ atica Pura e Aplicada, Estrada Dona Castorina 110, 22460- 320 Rio de Janeiro, RJ, Brazil. E-mail: amaury@impa.br † Havana University 1