Instances of finite-dimensional models, which are not the usual models of invariant systems evolving on the unitary groups, arising in problems of control and identification of quantum mechanical systems are given. Examples include problems quadratic in the creation and annihilation operators, problems where the control objective is to shape the Wigner distribution function, instances of problems with decoherence which can be formulated as invariant systems on non-unitary groups, control systems for the evolution of expectation values etc.