  We discuss the level-crossing field configurations for which the quantum
time-dependent two-state problem is solvable in terms of the confluent Heun
functions. We show that these configurations belong to fifteen four-parametric
families of models that generalize all the known 3- and 2-parametric families
for which the problem is solvable in terms of the Gauss hypergeometric and the
Kummer confluent hypergeometric functions. Analyzing the general case of
variable Rabi frequency and frequency detuning we mention that the most notable
features of the models provided by the derived classes are due to the extra
constant term in the detuning modulation function. Due to this term the classes
suggest numerous symmetric or asymmetric chirped pulses and a variety of models
with two crossings of the frequency resonance. The latter models are generated
by both real and complex transformations of the independent variable. In
general, the resulting detuning functions are asymmetric, the asymmetry being
controlled by the parameters of the detuning modulation function. In some
cases, however, the asymmetry may be additionally caused by the amplitude
modulation function. We present an example of the latter possibility and
additionally mention a constant amplitude model with periodically repeated
resonance-crossings. Finally, we discuss the excitation of a two-level atom by
a pulse of Lorentzian shape with a detuning providing one or two crossings of
the resonance. Using a series expansion of the solution of the confluent Heun
equation in terms of the Kummer hypergeometric functions we derive particular
closed form solutions of the two-state problem for this field configuration.
The particular sets of the involved parameters for which these solutions are
obtained define curves in the 3D space of the involved parameters belonging to
the complete return spectrum of the considered two-state quantum system.
