Summary: We consider two-dimensional maps generalizing the semistandard map by allowing more general analytic nonlinear terms having only Fourier components \(f_\nu\) with positive label \(\nu\), and study the analyticity properties of the function conjugating the motion on analytic homotopically nontrivial invariant curves to rotations. Then we show that, if the perturbation parameter is suitably rescaled, when the rotation number tends to a rational value nontangentially to the real axis from complex values, the limit of the conjugating function is a well defined analytic function. The rescaling depends not only on the limit value of the rotation number, but also on the map, and it is obtainable by the solution of a diophantine problem: so no universality property is exhibited. We also show that the rescaling can be different from that of the corresponding generalized standard maps, i.e. of the maps also having the Fourier components \(f_{-\nu}=f_\nu^*\). The results allow us to give quantitative bounds, from above and from below, on the radius of convergence of the limit function for generalized standard maps in the case of nonlinear terms which are trigonometric polynomials, solving a problem left open in a previous work of ours.