  We suggest that trialgebraic symmetries migth be a sensible starting point
for a notion of integrability for two dimensional spin systems. For a simple
trialgebraic symmetry we give an explicit condition in terms of matrices which
a Hamiltonian realizing such a symmetry has to satisfy and give an example of
such a Hamiltonian which realizes a trialgebra recently given by the authors in
another paper. Besides this, we also show that the same trialgebra can be
realized on a kind of Fock space of q-oscillators, i.e. the suggested
integrability concept gets via this symmetry a close connection to a kind of
noncommutative quantum field theory, paralleling the relation between the
integrability of spin chains and two dimensional conformal field theory.
