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Reduction for Painlevé equations at the fixed singular points of the second kind

This paper gives a simple reduction theorem for Painleve equations near the fixed singular points of the second kind in the framework of Hamiltonian mechanics. It is known that each Painleve equation P, (J=I, ... , VI) is equivalent to a Hamiltonian system : dA/dt=aH,/da, d~C/dt=-aH,/dA, where the Hamiltonian function H =-H,(t, A, p) is a polynomial of A and p of which the coefficients are rational functions of t ([14]). We call these Hamiltonian systems Painleve systems. Then the fixed singular points are formally classified as follows : a fixed singular point of a Painleve equation is o f the first kind or o f the second kind if Poincare rank of the corresponding Painleve system at the point is zero or positive respectively. We want to construct a 2-parameter family of solutions of each Painleve system at each fixed singular point, in other words, to obtain a local biholomorphic transformation which reduces it to a solvable system. As is well known, concerning the construction of an n-parameter family of solutions of an n-system at a fixed singular point, we have a general theorem by J. Malmquist under the so-called Poincare's condition ([12], [8]). However, we can not apply the theorem to Painleve systems because Poincare's condition is completely violated for them. Recently, having been stimulated by the idea of M. Iwano ([9]), several authors have obtained 2-parameter families of solutions of Painleve systems at the fixed singular points of the second kind ([16], [15], [19], [20]). Their works especially those by S. Yoshida explain, from a general point of view, the reason why the formal transformations for Painleve systems without