Abstract: In connection with non-Fuchsian equations Pnincare has made an important conclusion.It is impossible to obiain explicit expressions of irregular integrals To elucidate the essence of Pomcare’s problem, we establish correspondence theorem.Irregular inlegrals are analytic functions of new kind, possessing tree structure; part of which can be represented by conventional recursive series, while its remaining part is expressed by the so-called tree series, nol subjecting to any recursive relation at all.In contrast to the numerical solution calculated by infinite determinant of classical theory ( Hill-Poincare-von Koch), our method yields naturally exact analytic soiution in explicit form.The method proposed may be used lo construct a unifying theory for general equations wilh variahle coefficients, having various kinds of singularities as singular lines.The significance of Poincare conjecture is discussed, the tree series obtained belong to higher automorphic functions.

Key words: averaging discontinuous finite element, ultraconvergence, Hamiltonian system, momentum conservation