  We consider the problem of defining quantum integrability in systems with
finite number of energy levels starting from commuting matrices and construct
new general classes of such matrix models with a given number of commuting
partners. We argue that if the matrices depend on a (real) parameter, one can
define quantum integrability from this feature alone, leading to specific
results such as exact solvability, Poissonian energy level statistics and to
level crossings.
