  We explore superconducting instability for a clean two-band layered
superconductor with deep and shallow bands in the magnetic field applied
perpendicular to the layers. In the shallow band, the quasiclassical
approximation is not applicable, and Landau quantization has to be accounted
for exactly. The electronic spectrum of this band in the magnetic field is
composed of the one-dimensional Landau-level minibands. With increasing
magnetic field the system experiences series of Lifshitz transitions when the
chemical potential enters and exits the minibands. These transitions profoundly
influence the shape of the upper critical field at low temperatures. In
addition, the Zeeman spin splitting may cause the nonuniform state with
interlayer modulation of the superconducting order parameter
(Fulde-Ferrell-Larkin-Ovchinnikov state). Typically, the quantization effects
in the shallow band strongly promote the formation of this state. The uniform
state remains favorable only in the exceptional resonance cases when the
spin-splitting energy exactly matches the Landau-level spacing. Furthermore,
for specific relations between electronic spectrum parameters, the alternating
FFLO state may realize, in which the order parameter changes sign between the
neighboring layers. For all above cases, the reentrant high-field
superconducting states may emerge at low temperatures if the shallow band has
significant contribution to the Cooper pairing.
