The proof of existence of Vlasov-Maxwell equilibria which do not exhibit a functional dependence in terms of the single-particle energy is established. The theory deals with the kinetic treatment of multispecies axisymmetric magnetized plasmas, with particular reference to plasma systems which are slowly time varying. Aside from collisionless laboratory plasmas, the theory concerns important aspects of astrophysical scenarios, such as accretion-disk and coronal plasmas arising in the gravitational field of compact objects. Qualitative properties of the solution are investigated by making use of a perturbative kinetic theory. These concern the realization of the equilibrium kinetic distribution functions in terms of generalized Gaussian distributions and the constraints imposed by the Maxwell equations. These equilibria are shown to be generally non-neutral and characterized by the absence of the Debye screening effect. As a further application, the stability properties of these equilibria with respect to axisymmetric electromagnetic perturbations are addressed. This permits us to establish absolute stability criteria holding in such a case.