The Hesse coordinates for an algebraic cubic curve

An algebraic function given by the cubic equation is studied. The function is the spectral curve for a homogenous system of recurrence relations for sequences of $q$-polynomials, appearing in quantum field theory. To find the asymptotics basis of the solutions of these recurrence relations, we do parametrization of the spectral curve by means of the double covering of the cubic Hesse curve.