# Geometry of PDEs

One of the research lines focuses on some aspects of the geometry of PDEs. A classical problem is to classify metrics admitting at least one projective symmetry, i.e., a vector field whose local flow preserves the geodesic curves. Of these metrics it makes sense to classify both the projective class, which is formed by the metrics admitting the same geodesic curves (this class can be represented by a particular ODEs system), and the isometric one.This problem is a part of the more general one of constructing and classifying PDEs having a given symmetry group (transformations that send solutions into

solutions), which requires the study of the invariants of the aforementioned group. Another aspect of the research program is the study of integrable systems of both finite and infinite dimension. This study aims to characterize, in the case of systems of infinite dimension, those that are integrable in the solitonic sense. This can be done by studying the so-called fundamental algebras. In the case of integrable systems of finite dimension, the main objective is the study of the integrability of the geodetic flow, which is also connected to the existence of the aforementioned projective symmetries by virtue of the fact that one can always construct a quadratic integral in momenta starting from a projective symmetry.

** **