  Discussed is relationship between nonlinearity and symmetry of dynamical
models. The special stress is laid on essential, non-perturbative nonlinearity,
when none linear background does exist. This is nonlinearity essentially
different from ones given by nonlinear corrections imposed onto some linear
background. In a sense our ideas follow and develop those underlying
Born-Infeld electrodynamics and general relativity. We are particularly
interested in affine symmetry of degrees of freedom and dynamical models.
Discussed are mechanical geodetic models where the elastic dynamics of the body
is not encoded in potential energy but rather in affinely-invariant kinetic
energy, i.e., in affinely-invariant metric tensors on the configuration space.
In a sense this resembles the idea of Maupertuis variational principle. We
discuss also the dynamics of the field of linear frames, invariant under the
action of linear group of internal symmetries. It turns out that such models
have automatically the generalized Born-Infeld structure. This is some new
justification of Born-Infeld ideas. The suggested models may be applied in
nonlinear elasticity and in mechanics of relativistic continua with
microstructure. They provide also some alternative models of gravitation
theory. There exists also some interesting relationship with the theory of
nonlinear integrable lattices.
