  The integrable, (1+1) Gross-Pitaevskii (GP-) equation with hermitian property
is extended to chaotic behaviour as part of general complex fields within the
sl(2,C) algebra for Lax pairs. Furthermore, we prove the involution property of
conserved quantities in the case of GP-type equations with an arbitrary
external potential. We generalize the approach of Lax pair matrices to
arbitrary spacetime dimensions and conclude for the type of nonlinear equations
from the structure constants of the underlying algebra. One can also calculate
conserved quantities from loops within the (N-1) dimensional base space and the
mapping to the manifold of the general SL(n,C) group or a sub-group, provided
that the resulting fibre space is of nontrivial homotopic kind.
