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The problem of formulating a kinetic treatment for quasi-stationary collisionless plasmas in axisymmetric systems subject to the possibly independent presence of local strong velocity-shear and supersonic rotation velocities is posed. The theory is developed in the framework of the Vlasov-Maxwell description for multi-species non-relativistic plasmas. Applications to astrophysical accretion discs arising around compact objects and to plasmas in laboratory devices are considered. Explicit solutions for the equilibrium kinetic distribution function (KDF) are constructed based on the identification of the relevant particle adiabatic invariants. These are shown to be expressed in terms of generalized non-isotropic Gaussian distributions. A suitable perturbative theory is then developed which allows for the treatment of non-uniform strong velocity-shear/supersonic plasmas. This yields a series representation for the equilibrium KDF in which the leading-order term depends on both a finite se