  We present a specific constant-amplitude periodic level-crossing model of the
semi-classical quantum time-dependent two-state problem that belongs to a
general Heun class of field configurations. The exact analytic solution for the
probability amplitude, generally written for this class in terms of the general
Heun functions, in this specific case admits series expansion in terms of the
incomplete Beta functions. Terminating this series results in an infinite
hierarchy of finite-sum closed-form solutions each standing for a particular
two-state model, which generally is only conditionally integrable in the sense
that for these field configurations the amplitude and phase modulation
functions are not varied independently. However, there exists at least one
exception when the model is unconditionally integrable, that is the Rabi
frequency and the detuning of the driving optical field are controlled
independently. This is a constant-amplitude periodic level-crossing model, for
which the detuning in a limit becomes a Dirac delta-comb configuration with
variable frequency of the level-crossings. We derive the exact solution for
this model, determine the Floquet exponents and study the population dynamics
in the system for various regions of the input parameters.
