Free Probability for Pairs of Faces II: 2-Variables Bi-free\n $R$-Transform and Systems with Rank $\\le 1$ Commutation

We compute the generating series for the simplest class of bi-free cumulants,\nbeyond free cumulants, the two-bands bi-free cumulants of a pair of a left and\na right variable. We also consider two-faced systems with a commutation\ncondition implying that two-bands moments, that is expectation values of the\nproduct of a monomial of left and a monomial of right variables determine the\nother moments. Examples include hyponormal operators, dual systems in free\nentropy theory and bi-partite systems.\n