  The problem posed by the possible existence/non-existence of spatially
non-symmetric kinetic equilibria has remained unsolved in plasma theory. For
collisionless magnetized plasmas this involves the construction of stationary
solutions of the Vlasov-Maxwell equations. In this paper the issue is addressed
for non-relativistic plasmas both in astrophysical and laboratory contexts. The
treatment is based on a Lagrangian variational description of single-particle
dynamics. Starting point is a non-perturbative formulation of gyrokinetic
theory, which allows one to construct "a posteriori" with prescribed order of
accuracy an asymptotic representation for the magnetic moment. In terms of the
relevant particle adiabatic invariants generalized bi-Maxwellian equilibria are
proved to exist. These are shown to recover, under suitable assumptions, a
Chapman-Enskog form which permits an analytical treatment of the corresponding
fluid moments. In particular, the constrained posed by the Poisson and the
Ampere equations are analyzed, both for quasi-neutral and non-neutral plasmas.
The conditions of existence of the corresponding non-symmetric kinetic
equilibria are investigated. As a notable feature, both astrophysical and
laboratory plasmas are shown to exhibit, under suitable conditions, a kinetic
dynamo, whereby the equilibrium magnetic field can be self-generated by the
equilibrium plasma currents.
