  We will study two types of special solutions of the sixth Painleve equation,
which are invariant under the symmetries obtained from the Backlund
transformations. In most cases, the fixed points of the Backlund
transformations are classical solution, but our solutions are not classical for
generic parameters. We will calculate the linear monodromy of these solutions
exactly, and we will characterize them on Fricke's cubic surface of monodromy.
