In 1979, M. P. Drazin introduced $*$-regular semigroups as semigroups equipped with an involution in which every element has a generalized (Moore-Penrose) inverse. Since then, many characterizations of $*$-regular semigroups have been given, e.g. by Nambooripad and Pastijn, and others. In 1982, S. Crvenković described $*$-regular semigroups in terms of solvability of a certain type of linear equations in the involution semigroups. In this short note we prove that the solutions of all such equations are unique if and only if the binary reduct of the considered $*$-regular semigroup is a group.