we consider symmetric hopping $J_x^k = J_y^k = J$, identical chemical potential $\tilde{\mu}^k_{p,q} = \mu$ and $U_{kk} = U$. The phase diagram for $U_{12} =0$ is shown in Fig. \[ebhm\_v12\_0\] (a). It is identical to the phase diagram of single species extended BHM [@iskin_11; @kuldeep_19] and consists of the DW(1,0), MI(1,1), DW(2,1), MI(2,2), SS (green line) and SF phases. In the figure, the SS phase occurs as a thin envelope around the DW lobes. On increasing $V_k$ but keeping the other parameters fixed, the size of the DW lobes and the accompanying envelope of SS phase are enhanced. However, the MI lobes shrinks and disappear from the phase diagram. This is due to the higher energy cost of having commensurate occupancy due to the intra-species NN interaction. The same effect is reported in the single species eBHM [@iskin_11; @kuldeep_19].

The DW phases with $U_{12} =0$ are four-fold degenerate. Two of the states have $\Delta \rho^1=\Delta\rho^2$ and the other two have $\Delta \rho^1=-\Delta\rho^2$. For both set of states, one of the degenerate states is obtained by shifting both of the species by one lattice constant either along $x$ or $y$ direction. For the $\Delta \rho^1=\Delta\rho^2$ states, the occupancies of the two 