This paper is concerned with a class of non-symmetric operators, that is, π₯-symmetric operators, in Hilbert spaces. A sufficient condition for Ξ» β C being an element of the essential spectrum of a π₯-symmetric operator is given in terms of the number of linearly independent solutions of a certain homogeneous equation, and a characterization for points of the essential spectrum plus the set of all eigenvalues of a π₯-symmetric operator is obtained in terms of the numbers of linearly independent solutions of certain inhomogeneous equations. As direct applications, the corresponding results are obtained for singular π₯-symmetric Hamiltonian systems and their special forms of singular Sturm-Liouville equations with complex-valued coefficients, which enable us to study the spectra of singular π₯-symmetric differential expressions using numerous tools available in the fundamental theory of differential equations.