System of three-wave resonant interaction and $P$-type equations

Conditions of the existence of rational and one-parametrical families of solutions are obtained for the sixth Painleve equation. New integrability conditions in terms of the elliptic functions or superpositions of the elliptic and hypergeometrical functions and their derivatives are pointed out. With the help of these results exact classes of automodel solutions are constructed for a system of three-wave resonant interaction.