  Symmetries of the periodic Toda lattice are expresssed in action-angle
coordinates and characterized in terms of the periodic and Dirichlet spectrum
of the associated Jacobi matrices. Using these symmetries, the phase space of
the lattice with Dirichlet boundary conditions is embedded into the phase space
of a higher-dimensional periodic lattice. As an application, we obtain a
Birkhoff normal form and a KAM theorem for the lattice with Dirichlet boundary
conditions.
