In this work we investigate the resonant frequencies of composite metallic-anisotropic open resonators. The open resonator is formed by a perfect electric conductor (PEC) sphere coated by a gyroelectric spherical shell. We develop a surface-volume integral equation (SVIE) method, in conjunction with Dini series expansion. An orthogonal set of Dini-type spherical vector wave functions (SVWFs) is employed for the expansion of the field quantities, and allows the reduction of the SVIE to an homogeneous algebraic linear system. The validity of the developed method is tested by comparison to the separation of variables method (SVM), and numerical results are given for the normalized resonant wavenumbers and quality factors.