A minimal residual method, called MINRES-N2, that is based on the use of unconventional Krylov subspaces was previously proposed by the authors for solving a system of linear equations $Ax=b$ with a normal coefficient matrix whose spectrum belongs to an algebraic second-degree curve $\Gamma$. However, the computational scheme of this method does not cover matrices of the form $A=\alpha U+\beta I$, where $U$ is an arbitrary unitary matrix; for such matrices, $\Gamma$ is a circle. Systems of this type are repeatedly solved when the eigenvectors of a unitary matrix are calculated by inverse iteration. In this paper, a modification of MINRES-N2 suitable for linear polynomials in unitary matrices is proposed. Numerical results are presented demonstrating the significant superiority of the modified method over GMRES as applied to systems of this class.