  A procedure allowing for the construction of Lorentz invariant integrable
models living in d+1 dimensional space-time and with an n dimensional target
space is provided. Here, integrability is understood as the existence of the
generalized zero-curvature formulation and infinitely many conserved
quantities. A close relation between the Lagrange density of the integrable
models and the pullback of the pertinent volume form on target space is
established. Moreover, we show that the conserved currents are Noether currents
generated by the volume preserving diffeomorphisms. Further, we show how such
models may emerge via abelian projection of some gauge theories.
  Then we apply this framework to the construction of integrable models with
exotic textures. Particularly, we consider integrable models providing exact
suspended Hopf maps i.e., solitons with a nontrivial topological charge of
pi_4(S^3).
  Finally, some families of integrable models with solitons of pi_n(S^n) type
are constructed. Infinitely many exact solutions with arbitrary value of the
topological index are found. In addition, we demonstrate that they saturate a
Bogomolny bound.
