  Within the Landau-Ginsburg-Devonshire phenomenological approach we study the
ferroic nanosystems properties changes caused by the flexo-effect
(flexoelectric, flexomagnetic, flexoelastic) existing spontaneously due to the
inhomogeneity of order parameters. Exact solution for the spatially
inhomogeneous mechanical displacement vector allowing for flexo-coupling
contribution was found for nanowires and thin pills. Strong influence of
flexo-effect in nanorods and thin pills leads to the displacements of the atoms
resulting into the unit cell symmetry changes, which lead to the phase
transition temperature shift, as well as the flat geometry in radial direction
transforms into the saucer-like one. The new phenomena can be considered as
true manifestation of the spontaneous flexo-effect existence. It was shown that
flexo-effect leads to (a) the appearance of new linear and nonlinear
contribution and renormalization of coefficients before the order parameter
gradient, (b) essentially influences the transition temperature, piezoelectric
response and the spatial distribution of the order parameter, (c) results in
renormalization of extrapolation length in the boundary conditions.
