A family of the so-called Maxwell times which arises in the contexto of Higher-Order Generalized Hydrodynamics (also called Mesoscopic Hydro-Thermodynamics) is evidenced. This is done in the framework of a HOGH build within a statistical foundation i
n terms of a Non-Equilibrium Statistical Ensemble Formalism. It consists in a description in terms of the densities of particles and energy and their fluxes of all orders, with the motion described by a set of coupled nonlinear integro-differential equations involving them. These Maxwell Times have a fundamental role in determining the type of hydrodynamic motion that the system would display in the given condition and constraints. The different types of motion are well described by contractions of the full description done in terms of a reduced number of fluxes up to a certain order.
