  Using shortcuts to adiabaticity, we solve the time-dependent Schroedinger
equation that is reduced to a classical nonlinear integrable equation. For a
given time-dependent Hamiltonian, the counterdiabatic term is introduced to
prevent nonadiabatic transitions. Using the fact that the equation for the
dynamical invariant is equivalent to the Lax equation in nonlinear integrable
systems, we obtain the counterdiabatic term exactly. The counterdiabatic term
is available when the corresponding Lax pair exists and the solvable systems
are classified in a unified and systematic way. Multisoliton potentials
obtained from the Korteweg-de Vries equation and isotropic XY spin chains from
the Toda equations are studied in detail.
