Class of invariants for the 2D time-dependent Landau problem and harmonic oscillator in a magnetic field

We consider an isotropic two dimensional harmonic oscillator with arbitrarily time-dependent mass $M(t)$ and frequency $\mathrm{\Omega }(t)$ in an arbitrarily time-dependent magnetic field $B(t)$. We determine two commuting invariant observables (in the sense of Lewis and Riesenfeld) $L,I$ in terms of some solution of an auxiliary ordinary differential equation and an orthonormal basis of the Hilbert space consisting of joint eigenvectors ${\varphi }_{\lambda }$ of $L,I$. We then determine time-dependent phases ${\alpha }_{\lambda }(t)$ such that the ${\psi }_{\lambda }(t)=e^{i{\alpha }_{\lambda }}{\varphi }_{\lambda }$ are solutions of the time-dependent Schr"odinger equation and make up an orthonormal basis of the Hilbert space. These results apply, in particular to a two dimensional Landau problem with time-dependent $M,B$, which is obtained from the above just by setting $\mathrm{\Omega }(t)\equiv 0$. By a mere redefinition of the parameters, these results can be applied also to the analogous models on the canonical non-commutative plane.