The motion of an elastic composite shell over a hard rough surface in the presence of combined anisotropic dry friction is considered. This model can be used to study the dynamics of pneumatics (aviation and automotive) in conditions of combined kinematics, as well as various control robotics systems. To correctly account for the influence of the anisotropy of dry friction coefficients in such systems, it is required to construct approximate analytical models of the force state inside the contact spot, taking into account the real distribution of normal and tangential contact stresses. The contact pressure distribution is constructed using the S.A. Ambartsumyan for a transversally isotropic spherical shell. This equation is modified by introducing additional relationships for the reduced contact pressure and normal displacements. The construction of the resolving integral equation for the contact pressure is based on the principle of superposition and the method of Green’s functions. For this, the corresponding Green’s function is constructed, which is the normal displacement of the shell as a solution to the problem of the effect of concentrated pressure. Green’s function as well as the contact pressure, it is sought in the form of series expansions in Legendre polynomials, taking into account additional relations for the reduced contact pressure and normal displacements. Using the Green’s function, an integral equation solving the problem is constructed. As a result, the problem is reduced to determining the expansion coefficients in a series of the reduced contact pressure. Restricting ourselves to a finite number of terms in the series of expansions, using the discretization of the contact area and the properties of Legendre polynomials, the problem is reduced to solving a system of algebraic equations for the expansion coefficients for the reduced pressure. After that, from the additional relation, the coefficients of the required expansion of the contact pressure in a series in Legendre polynomials are determined.