As rigorous methodology for studying the Riemann-Hilbert problems associated with certain integrable nonlinear ODEs was introduced in 1992 by Fokas and Zhou, and was used to investigate Painleve II and Painleve IV equations. Here the authors apply this methodology to Painleve I, III, and V equations. They show that the Cauchy problems for these equations admit in general global solutions, meromorphic in t. Furthermore, for special relations among the monodromy data and for t on Stokes lines, these solutions are bounded for finite t. In connection with Painleve I they note that the usual Lax pair gives rise to monodromy data some of which depend nonlinearly on the unknown solution of Painleve I. This problem is bypassed here by introducing a new Lax pair for which all the monodromy data are constant.

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A method to obtain the Schlesinger transformations for Painlevi VI equation is given. The procedure involves formulating a Riemann-Hilbert problem for a transformation matrix which transforms the solution of the linear problem but leaves the associated monodromy data the same.

Starting from the first Painlevé equation, Painlevé type equations of higher order are obtained by using the singular point analysis. 4 j=0 c j (z)y j = 0, (1.3)

The algorithmic method introduced by Fokas and Ablowitz to investigate the transformation properties of Painleve equations is used to obtain a one-to-one correspondence´ between the Painleve I, II and III equations and certain second-order second degree equations´ of Painleve type.´