Full self-consistent stationary Vlasov-Maxwell solutions of magnetically confined plasmas are built for systems with cylindrical symmetries. The stationary solutions are thermodynamic equilibrium solutions. These are obtained by computing the equilibrium distribution function resulting from maximizing the entropy and closing the equations with source terms that are then computed by using the obtained distribution. This leads to a self-consistent problem corresponding to solving a set of two coupled second order nonlinear differential equations. Relevant plasma parameters are introduced and a bifurcation leading to an improvement of plasma confinement is shown. Conversely, in the improved confinement setting, we exhibit the emergence of a separatrix in the integrable motion of a charged particle .