Seminarium "The Trans-Carpathian Seminar on Geometry & Physics"

A Lie system is a nonautonomous system of first-order ordinary differential equations possessing a superposition rule, namely a mappingallowing us to describe its general solution in terms of a generic finitefamily of particular solutions and a set of constants. Equivalently, we cancharacterise Lie systems as systems of first-order ordinary differentialequations describing the integral curves of a time-dependent vector fieldtaking values in a finite-dimensional Lie algebra of vector fields: a Vessiot--Guldberg Lie algebra.We introduce a new class of Lie systems possessing a Vessiot--Guldberg Liealgebra of Hamiltonian vector fields with respect to a Dirac structure: the Dirac--Lie systems.The use of Dirac geometry enable us to develop powerful methods to studythe constants of motion, superposition rules, and other properties of thesesystems.Our results generalise previous methods to investigate integrable systemsand certain types of Lie systems.We illustrate our findings with the study of several types of Schwarzianequations and other differential equations of interest.