  We consider the model of two coupled oscillators with Kerr nonlinearities in
the rotating-wave approximation. We demonstrate that for a certain set of
parameters of the model, the multi-photon transitions occur between many pairs
of the oscillator states simultaneously. Also, the position of the multi-photon
resonances does not depend on the coupling strength between two oscillators. We
prove rigorously that this is a consequence of a certain symmetry of the
perturbation theory series for the model. In addition, we analyse the model in
the quasi-classical limit by considering the dynamics of the pseudo-angular
momentum. We identify the multi-photon transitions with the tunnelling
transitions between the degenerate classical trajectories on the Bloch sphere.
