Abstract <p><i>Background.</i> Conjugate boundary value problems for a system of Maxwell’s equations are fundamental in electrodynamics. Recently, there has been interest in the problems involving the presence of a thin graphene layer on the surface, which changes the conjugation conditions. The aim of this study is to obtain a system of integral equations for the vector boundary value problem on electromagnetic oscillations of a graphene-coated dielectric ball in a spherical coordinate system. <i>Materials and methods</i>. Using the Stratton‒Chu formula, the components of the field are expressed through surface currents and the conjugation conditions are used to obtain singular integral equations on the ball surface. <i>Results.</i> A system of singular integral equations with four unknown scalar functions on the sphere surface is derived. The properties of the system of integral equations are studied. <i>Conclusions</i>. The resulting system of singular integral equations can be solved numerically by known methods, for example, the Galerkin method or the collocation method.</p>