  Classical string actions in AdS(3) and dS(3) can be connected to the
sinh-Gordon and cosh-Gordon equations through Pohlmeyer reduction. We show that
the problem of constructing a classical string solution with a given static or
translationally invariant Pohlmeyer counterpart is equivalent to solving four
pairs of effective Schrodinger problems. Each pair consists of a flat potential
and an n = 1 Lame potential whose eigenvalues are connected, and, additionally,
the four solutions satisfy a set of constraints. An approach for solving this
system is developed by employing an interesting connection between the specific
class of classical string solutions and the band structure of the Lame
potential. This method is used for the construction of several families of
classical string solutions, one of which turns out to be the spiky strings in
AdS(3). New solutions include circular rotating strings in AdS(3) with singular
time evolution of their radius and angular velocity as well as classical string
solutions in dS(3).
