The paper presents an analytic solution to the problem on stability and initial post-buckling behavior of a cylindrical shell made from anisotropic material with one plane of elastic symmetry and loaded by torsion moments on the ends. Solution is based on the equations taken from the Donnel-Mushtary-Vlasov theory of anisotropic shells and linearized by the methods of the Koiter theory. Main stage of the problem solution is reduced to systems of algebraic equations which are then analyzed.