  We derive 35 five-parametric classes of the quantum time-dependent two-state
models solvable in terms of the general Heun functions. Each of the classes is
defined by a pair of generating functions the first of which is referred to as
the amplitude- and the second one as the detuning-modulation function. The
classes suggest numerous families of specific field configurations with
different physical properties generated by appropriate choices of the
transformation of the independent variable, real or complex. There are many
families of models with constant detuning or constant amplitude, numerous
classes of chirped pulses of controllable amplitude and/or detuning, families
of models with double or multiple (periodic) crossings, periodic amplitude
modulation field configurations, etc. We present several families of
constant-detuning field configurations the members of which are symmetric or
asymmetric two-peak finite-area pulses with controllable distance between the
peaks and controllable amplitude of each of the peaks. We show that the edge
shapes, the distance between the peaks as well as the amplitude of the peaks
are controlled almost independently, by different parameters. We identify the
parameters controlling each of the mentioned features and discuss other basic
properties of pulse shapes. We show that the pulse edges may become step-wise
functions and determine the positions of the limiting vertical-wall edges. We
show that the pulse width is controlled by only two of the involved parameters.
For some values of these parameters the pulse width diverges and for some other
values the pulses become infinitely narrow. We show that the effect of the two
mentioned parameters is almost similar, that is, both parameters are able to
independently produce pulses of almost the same shape and width.
