  We describe some new exact solutions for two- and four-level systems. In all
the cases, external fields have a restricted behavior in time. First, we
consider two types of new solutions for one-spin equation, one of them is in a
external magnetic field that acts during a finite time interval. A new solution
for two interacting spins is found in the case when the field difference
between the external fields in each spin vary adiabatically, vanishing on the
time infinity. The latter system can be identified with a quantum gate realized
by two coupled quantum dots. The probability of the Swap operation for such a
gate can be explicitly expressed in terms of special functions. Using the
obtained expressions, we construct plots for the Swap operation for some
parameters of the external magnetic field and interaction function.
