A new approach to solve the problem of light scattering by particles with an axial symmetry is suggested. We divide the electromagnetic fields in two parts (an axisymmetric part having no dependence on the azimuthal angle and a non-axisymmetric part whose averaging over this angle gives zero) and consider the scattering problem separately for each of the two parts. Another feature of our approach is the special choice of the scalar potentials. The potentials connected with the azimuthal components of the electromagnetic fields are used for the axisymmetric parts of these fields, and a superposition of the Debye potentials and the vertical components of the Hertz vector is utilized for the non-axisymmetric parts. The scattering problem is formulated in the form of integral equations. The scalar potentials are expanded in terms of the spherical wave functions, and the expansion coefficients are determined from a solution of the infinite systems of linear algebraic equations. A numerical code based on the approach is developed, and test results obtained for spheroids are presented.