Singular ordinary differential equations homogeneous of degree 0 near a codimension 2 set

This work deals with an example of class of ordinary differential equations which are singular near a codimension 2 set, with an homogeneous singularity of degree 0. Under some structural assumptions, we prove that for almost all initial data there exists a unique global solution and study the evolution of the Lebesgue measure of a transported set of initial data. The analysis is motivated by the Low Mach number asymptotics of compressible fluid models in the case of non isentropic flows, which involves such dynamical systems