  In this thesis, we investigate various integral submodels and generalize
them. In part I, we study the submodel of the nonlinear $\mathbf{C}P^1$-model
and the related submodels in $(1+2)$ dimensions. In part II, we construct
integrable submodels of the nonlinear Grassmann models in any dimension. We
call them the Grassmann submodels. To show that our submodels are integrable,
we construct an infinite number of conserved currents in two ways. One is that
we make full use of the Noether currents of the nonlinear Grassmann models. The
other is that we use a method of multiplier. Next we investigate symmetries of
the Grassmann submodel. By using the symmetries, we can construct a wide class
of exact solutions for our submodels. In part III, keeping some properties of
our submodels, we generalize our submodels to higher-order equations. First we
prepare the Bell polynomials and the generalized Bell polynomials which play
the most important roles in our theory of generalized submodels. Next we
generalize the $\mathbf{C}P^1$-submodel to higher-order equations. Lastly we
generalize the Grassmann submodel to higher-order equations. By using the
generalized Bell polynomials, we can show that the generalized Grassmann
submodels are also integrable. As a result, we obtain a hierarchy of systems of
integrable equations in any dimension which includes Grassmann submodels. These
results lead to the conclusion that the integrable structures of our
generalized submodels are closely related to some fundamental properties of the
Bell polynomials.
