  The classical Euler's problem on stationary configurations of elastic rod
with fixed endpoints and tangents at the endpoints is considered as a
left-invariant optimal control problem on the group of motions of a
two-dimensional plane $\E(2)$.
  The attainable set is described, existence and boundedness of optimal
controls are proved. Extremals are parametrized by Jacobi's elliptic functions
of natural coordinates induced by the flow of the mathematical pendulum on
fibers of the cotangent bundle of $\E(2)$.
  The group of discrete symmetries of Euler's problem generated by reflections
in the phase space of the pendulum is studied. The corresponding Maxwell points
are completely described via the study of fixed points of this group. As a
consequence, an upper bound on cut points in Euler's problem is obtained.
