Adsorption of single gases and their binary mixture on a strongly heterogeneous surface is modeled using the integral equation (IE) approach. The adsorbing surface is assumed to be characterized by a rectangular distribution of the adsorption energy with respect to each component of the mixture. The influence of correlations between the distribution functions associated with different components on the corresponding adsorption isotherms and phase diagrams is discussed. The validity of the condensation approximation, which may be also applied to the model, is assessed by comparing the corresponding results with exact solutions of the IE obtained for single, as well as for mixed, adsorption. In addition, predictions of the theory are compared with the results of the grand canonical Monte Carlo (GCMC) simulations carried out for the system studied. it is shown that, in general, the IE approach gives the results that are very close to the simulation data, especially when the distribution functions have the same shape, that is, when they are highly correlated.