  The review is focused on our recent predictions of nontrivial physical
phenomena taking place in the nanostructure single-layer grapheme on
ferroelectric substrate, which are related with magnetic field. In particular
we predicted that 180-degree domain walls in a strained ferroelectric film can
induce p-n junctions in a graphene channel and lead to the unusual temperature
and gate voltage dependences of the perpendicular modes {\nu} of the integer
quantum Hall effect. The non-integer numbers and their irregular sequence
principally differ from the conventional sequence {\nu}= 3/2, 5/3. The unusual
v-numbers originate from significantly different numbers of the edge modes,
{\nu}1 and {\nu}2, corresponding to different concentration of carriers in the
left (n1) and right (n2) ferroelectric domains of p-n junction boundary. The
difference between n1 and n2 disappears with the vanishing of the film
spontaneous polarization in a paraelectric phase, which can be varied in a wide
temperature range by an appropriate choice of misfit strain originated from the
film-substrate lattice mismatch. Next we studied the electric conductivity of
the system ferromagnetic dielectric - graphene channel - ferroelectric
substrate. The magnetic dielectric locally transforms the band spectrum of
graphene by inducing an energy gap in it and making it spin-asymmetric with
respect to the free electrons. It was demonstrated, that if the Fermi level in
the graphene channel belongs to energy intervals, where the graphene band
spectrum, modified by EuO, becomes sharply spin-asymmetric, such a device can
be an ideal non-volatile spin filter. The practical application of the system
under consideration would be restricted by a Curie temperature of a
ferromagnet. Controlling of the Fermi level (e.g. by temperature that changes
ferroelectric polarization) can convert a spin filter to a spin valve.
