A new approach to solve the problem of light scattering by particles with an axial symmetry is developed. The electromagnetic fields are divided in two (axisymmetric and non-axisymmetric) parts. The scattering problem formulated in the integral form is solved separately for each of the parts using specially chosen scalar potentials. The potentials are expanded in terms of the spherical wave functions, and the expansion coefficients are determined from solution of the infinite systems of linear algebraic equations. A numerical code based on the approach is created and tested for spheroids, finite size cylinders, and Chebyshev particles.