Topology of first integrals via Milnor fibrations II

Abstract

This survey is the continuation of a series of works aimed at applying tools from Singularity Theory to Differential Equations. More precisely, we utilize the powerfull Milnor's Fibration Theory to give geometric-topological classifications of first integrals of differential systems. In the previous paper, systems of first-order quasilinear partial differential equations were examined, focusing on the case of an isolated singularity. Now, we address both cases of isolated and \textit{non-isolated singularities} for more general dynamical systems (namely, \textit{foliations}) that admit at least one first integral. For this, we utilize recently established connections between harmonic morphisms and Milnor fibrations to provide topological and geometric descriptions of the foliations under consideration. In particular, we apply these results to analyze the graph of solutions of some quasilinear systems.