The paper studies nonlinear mappings using methods of p-regularity theory and some concepts and technics of set-valued analysis. The main result addresses to the problem of existence of solutions to nonlinear equations in the degenerate case where a linear part may be singular at the considered initial point.

The paper presents the continuation of the previous results devoted to the problem of solutions existence to nonlinear equations in singular case where a linear part of considered mapping determining the equation may be degenerate at the corresponding initial point. We study the case when the p-kernel of the mapping is non trivial. Such type of problems appears in various mathematical models and applications. The p-regularity theory is used in our analysis and some concepts and technics of set-valued approach.

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Klein's Erlangen program contains the postulate to study the group of automorphisms instead of a structure itself. This postulate, taken literally, sometimes means a substantial loss of information. For example, the group of automorphisms of the field of rational numbers is trivial. However in the case of Euclidean plane geometry the situation is different. We shall prove that the plane Euclidean geometry is mutually interpretable with the elementary theory of the group of authomorphisms of its standard model. Thus both theories differ practically in the language only.