conditions for controllability that are generic [@PriSig10; @MasonSigalotti2010; @nersesyan] even in the case $p=1$. Nevertheless, in many examples interesting for applications these conditions cannot be directly applied or controllability fails to hold, as a consequence of the symmetries of the system. Symmetries can induce degeneracies in the spectrum (e.g. multiple eigenvalues or presence of equal spectral gaps) and reduce the coupling of eigenstates via the control. This happens, for instance, in a planar rotating molecule controlled by only one control [@ancoranoi Section 8] which is not (approximately) controllable.

The use of more than one control opens new controllability horizons.

Controllability results with more than one input have been obtained for specific systems [@ervedoza_puel; @brockettetc] and some general approximate controllability results between eigenfunctions have been proved via adiabatic methods [@Boscain_Adami; @adiabatiko]. The first multi-input result via Lie-algebraic methods is given in [@ancoranoi Section 8] where the problem of the spectral degeneracies in the planar rotating molecule has been overcome associating with every $1$-dimensional slice of the set of admissible controls an invariant subspace of the state space ${\mathcal{H}}$ on which the single-input controllability result applies. Anyhow, such a technique does not apply for more general cases. In the case of a 