the main result of Ref. [@Amirkhanov1966] but also the generalization of their conclusion for the case of nonidentical barriers ($V_1 \neq V_2$).

Second, Eq. (\[W\_equivalent\_particles\]) basically provides an explanation of the observed anomalies related to the potentials $\Omega_n (\alpha; x_1, x_2)$ \[Eq. (\[PotentialDeff\])\] pictured in Fig. \[Fig\_potentials\], if we recall that tunneling of a 2D particle in the potential $\Omega_n (\alpha; x_1, x_2)$ is equivalent to collective tunneling of two equal 1D particles through the potential barriers $V_1(x_1) = \alpha V(x_1)$ and $V_2(x_2) = 3 V(x_2)$. Hence, Eq. (\[SymmetryW\]) determines the selection rule for transitions between states of the internal degree of freedom induced by (collective) tunneling. The key point is that the probability of collective tunneling strongly depends on whether an excitation of the internal degree of freedom is possible. If a system is initially in the ground state and the excitations are allowed, then by going to an excited state, the center of mass of the system lowers its kinetic energy (i.e., increasing the width of the barrier), consequently reducing the probability of tunneling (see Refs. [@Zakhariev2002; @Goodvin2005; @Goodvin2005a; @Hnybida2008; @Shegelski2008] and reference therein). We recall that the parity of the ground state is even, the first excited 