Simultaneous local normal forms of dynamical systems with singular underlying geometric structures

The aim of this paper is to develop, for the first time, a general theory of simultaneous local normalisation of couples (X,G), where X is a dynamical system (vector field) and G is an underlying geometric structure preserved by X, even if both have singularities. Such couples appear naturally in many problems, e.g. Hamiltonian dynamics, where G is a symplectic structure and one has the theory of Birkhoff normal forms, or constrained dynamics, where G is a smooth, in general singular, distribution of tangent subspaces, etc. In this paper, the geometric structure G is of the following types: volume form, symplectic form, contact form, Poisson tensor, as well as their singular versions. The paper addresses mainly the more difficult situations when both X and G are singular at a point and its results prove the existence of natural simultaneous normal forms in these cases. In general, the normalisation is only formal, but when G and X are (real or complex) analytic and X is analytically or Darboux integrable, then the simultaneous normalisation is also analytic. Our theory is based on a new approach, called the Toric Conservation Principle, as well as the classical step-by-step normalisation technique, and the equivariant path method.