The authors define an orthogenerator for the deformation \(\Phi\), and show that the deformation \(\Phi '=\Phi -L(w)\) is special if w is an orthogenerator for \(\Phi\), where L(v) is the Schouten brackets between \(\Psi\) and v. Hence, it is possible to construct a generator of the deformation. Main results:
1. Let H be a smooth real function on Poisson manifold N, and \(\Psi\) a morphism of linear bundles \(\Phi\), then by a certain transformation exp(\(\epsilon\) v), with accuracy \(O(\epsilon^ 2)\), local passage from system \({\dot \xi}=\Psi (\xi)dH(\xi)+\epsilon \Phi (\xi)dH(\xi)\quad,\xi \in N,\) to the Hamiltonian system \({\dot \zeta}=\Psi (\zeta)dH\epsilon (\zeta),\quad H\epsilon =H+\epsilon v(H)\) is possible. When a generator v of \(\Phi\) exists globally if and only if the element in \(H^ 2_ 0(\Omega)\) corresponding to the cohomology class of the form \(\beta^ w\) is zero (here w is an orthogenerator of \(\Phi)\); we have \(v=\Psi \gamma^ w+w\), and it is unique to within addition to w of an arbitrary field in \(V_ 0(N)\). Where the orthogenerator w is chosen so that the form \(\beta^ w\) is exact, \(\gamma^ w\) is an antiderivative of this form: \(d\gamma^ w=\beta^ w.\)
2. If the restriction of \(\beta^ w\) to each Lagrangian submanifold \(\Delta\) is exact, i.e., it exists a 1-form \(\gamma_{\Lambda}\) on \(\Lambda\) such that \(\beta^ w|_{\Lambda}=d\gamma_{\Lambda}\) and \(\mu_{\Lambda}=\lim_{T\to \infty}[1/T\int^{1}_{0} \exp (\tau ad(H))M_{\Lambda}d\tau]\) is constant on \(\Lambda\), then the series of numbers \(\lambda_ m=H|_{\Lambda m}+\epsilon \mu_{\lambda m}\) is at a distance \(O(\epsilon^ 2+h^ 2)\) from the spectrum of H(A), where \(M_{\Lambda}=w(H)|_{\Lambda}-\gamma_{\Lambda}(ad(H))\) on submanifold \(\Lambda\), \(\Lambda_ m\) are obtained from the general scheme worked out by the second author and V. P. Maslov in Usp. Mat. Nauk 39, No.6(240), 115-173 (1984; Zbl 0588.58031).