This book discusses lots of different types of integrable models, both classical and quantum. As far as I'm aware this is the best intro book.
http://www.cambridge.org/ie/academi...e-scattering-method-and-correlation-functions
Now, it's really only spin chains I'm familiar with, but in this case the conserved quantities are given by taking the trace of the monodromy matrix on the auxiliary space. This defines the transfer matrix which generates a tower of commuting charges, one of which the is the Hamiltonian of the system. The one-dimensional spin chain with L sites and SU(2) symmetry has L degrees of freedom, whereas the transfer matrix only gives you L-1 conserved quantities. By adding a component of spin, say ##S^z##, we obtain the full set of commuting charges, and the system is integrable.