  We present a systematic, analytical study of geometrically thin, optically
thick accretion disc solutions for magnetized turbulent flows, with an
alpha-like viscosity prescription. Under the only assumptions that (1)
Magneto-Rotational instability (MRI) generates the turbulence that produces the
anomalous viscosity needed for accretion to proceed, and that (2) the magnetic
field amplified by the instability saturates due to buoyant vertical escape, we
are able to self-consistently solve the disc structure equations including the
fraction of power, f, that is carried off by vertical Poynting flux (and likely
dissipated outside the optically thick disc). For low-viscosity discs, we
obtain stable high-f solutions at low accretion rates, when gas pressure
dominates, and unstable, low-f, radiation pressure dominated solutions at high
accretion rates. For high viscosity discs, instead, a new thermally and
viscously stable, radiation pressure dominated solution is found, characterized
by f~1 and appearing only above a critical accretion rate (of the order of few
tenths of the Eddington one). We discuss the regimes of validity of our
assumptions, and the astrophysical relevance of our solutions. We conclude that
our newly discovered thin disc solutions, possibly accompanied by powerful,
magnetically dominated coronae and outflows, should be seriously considered as
models for black holes accreting at super-Eddington rates.
