The paper proposes the methods and a software package for solving systems of Boolean equations, the propositions of which are formulas over an arbitrary basis, Zhegalkin polynomials consisting of nonlinear components and disjunctive normal forms.The formation of Zhegalkin polynomials from formulas over an arbitrary basis, the finding of the maximum joint subsystems of logical equations and the search for intersections of the sets of solutions of several subsystems of the system of equations are considered.