For three-dimensional many-electron time periodic Hamiltonians which a
re invariant under dynamical symmetry of order N we prove that only th
e (nN +/- 1)th, n = 1,2,..., harmonics are generated. We discuss the a
pplication of the dynamical symmetry based selection rules to the gene
ration of high harmonics by thin crystals. The derived selection rules
are demonstrated numerically for a one-dimensional model, showing tha
t the dynamically symmetric systems can be used not only as ''filters'
' of the very high harmonics but also as their ''amplifiers.''