  The general semiclassical time-dependent two-state problem is considered for
a specific field configuration referred to as the generalized Rosen-Zener
model. This is a rich family of pulse amplitude- and phase-modulation functions
describing both non-crossing and term-crossing models with one or two crossing
points. The model includes the original constant-detuning non-crossing
Rosen-Zener model as a particular case. We show that the governing system of
equations is reduced to a confluent Heun equation. When inspecting the
conditions for returning the system to the initial state at the end of the
interaction with the field, we reformulate the problem as an eigenvalue problem
for the peak Rabi frequency and apply the Rayleigh-Schr\"odinger perturbation
theory. Further, we develop a generalized approach for finding the higher-order
approximations, which is applicable for the whole variation region of the
involved input parameters of the system. We examine the general surface in the
3D space of input parameters, which defines the position of the n-th order
return-resonance, and show that the section of the general surface is
accurately approximated by an ellipse. We find a highly accurate analytic
description through the zeros of a Kummer confluent hypergeometric function.
From the point of view of the generality, the analytical description of
mentioned curve for the whole variation range of all involved parameters is the
main result of the present paper.
