We will study the symmetric solutions of the Painlev\'e equations.
(i) We will study the symmetric solutions of P6 which are invariant under some replacements obtained from the B\"acklund transformations. In general, the fixed points of the B\"acklund transformations are Umemura's classical solutions. However, our solutions are not classical in general. We will caluclate the linear monodromy of these solutions exactry, and we will characterize them on the cubic surface of the monodromy. (joint work with K. Kaneko(Osaka Univ.))
(ii) We will study the Painlev\'e equation on finite field. For P1 and P2, we will show the relation between the solutions on finite field and the symmetric solutions. (joint work with K. Okamoto(Tokyo Univ.))