Rational Solutions of Painlev´ e Equations

Consider the sixth Painlevequation (P6) below where�,�, andare complex parame- ters. We prove the necessary and sufficient conditions for the existence of rational solutions of equa- tion (P6) in term of special relations among the parameters. The number of distinct rational solutions in each case is exactly one or two or infinite. And each of them may be generatedby means of transfor- mation group found by Okamoto (7) and Btransformations found by Fokas and Yortsos (4). Alistofrationalsolutionsisincluded intheappendix. Forthesakeofcompleteness,wecollectedallthe corresponding results of other five Painlevequations (P1)-(P5) below, which have been investigated by many authors (1)-(7).