Compatibility conditions for a membrane taking on the shape of a surface with given non-constant Gaussian curvature K are deduced. The system for the unknown displacement in general coordinates is integrated, by making use of the non-holonomic basis formed by the unit vectors tangent to the lines K=const. and to the associated orthogonal trajectories. This method unifies the discussion for rotation surfaces, geodesically parallel surfaces and general surfaces. In particular it is proved that, in the case of geodesically parallel surfaces, the integrability conditions reduce to two and involve explicitly the strain and its derivatives up to the third and fifth order respectively.