  The Fermi-Pasta-Ulam (FPU) lattice with periodic boundary conditions and $n$
particles admits a large group of discrete symmetries. The fixed point sets of
these symmetries naturally form invariant symplectic manifolds that are
investigated in this short note. For each $k$ dividing $n$ we find $k$ degree
of freedom invariant manifolds. They represent short wavelength solutions
composed of $k$ Fourier-modes and can be interpreted as embedded lattices with
periodic boundary conditions and only $k$ particles. Inside these invariant
manifolds other invariant structures and exact solutions are found which
represent for instance periodic and quasi-periodic solutions and standing and
traveling waves. Some of these results have been found previously by other
authors via a study of mode coupling coefficients and recently also by
investigating `bushes of normal modes'. The method of this paper is similar to
the latter method and much more systematic than the former. We arrive at
previously unknown results without any difficult computations. It is shown
moreover that similar invariant manifolds exist also in the Klein-Gordon
lattice and in the thermodynamic and continuum limits.
