  The linear instability of thin, vertically-isothermal Keplerian discs, under
the influence of axial magnetic field is investigated. Solutions of the
stability problem are found explicitly by asymptotic expansions in the small
aspect ratio of the disc. It is shown that the perturbations are decoupled into
in-plane and vertical modes. Exact expressions for the growth rates as well as
the number of unstable modes are derived. Those are the discrete counterpart of
the continuous infinite homogeneous cylinder magnetorotational (MRI) spectrum.
In addition, a weakly nonlinear analysis of the MRI is performed. It is shown
that near the instability threshold the latter is saturated by the stable
magnetoacoustic modes.
