There are many interesting results and open questions regarding the degree of regularity of Diophantine equations. The Fox and Kleitman conjecture (2006) concerning the maximum degree of regularity of the equation x_1 + • • • + x_k − y_1 − • • • − y_k = b_k , as b_k runs over the positive integers, has recently been confirmed (2017). We here consider the problem of finding the degree of regularity of some specializations of the above equation.