For each Painlevé system, we have a manifold, called the defining manifold, on which the system defines a uniform foliation. In this paper, we describe confluence processes in these manifolds as deformations of manifolds compatible to those in Painlevé systems. 1. Introduction. The purpose of this paper is to describe confluence processes in the defining manifolds for Painlevé systems, namely, to show a hierarchy of these manifolds. Each Painlevé equation x and y whose coefficients are rational

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... functions of t. Thus the Hamiltonian system (H J ) is called the J-th Painlevé system. For each Painlevé system, there is a manifold E J , called the defining manifold for the J-th Painlevé system, on which the system defines a uniform foliation. The manifold E J is a fiber space over the t-space B J = P − {the fixed singular points} (P denotes the complex projective line), containing as a fiber subspace the trivial fiber space C 2 × B J ( (x, y, t)) on which the system (H J ) defines a nonsingular foliation. It should be noted that the foliation on C 2 × B J ( (x, y, t)) is not uniform because the solution of (H J ) may have movable singularities, but that on E J is uniform, namely, (i) every leaf is transversal to fibers, (ii) for every P 0 ∈ E J , any curve in B J with the starting point π J (P 0 ) (π J denotes the projection from E J onto B J ) can be lifted on the leaf passing through the point P 0 ([4]). Each fiber E J (t) over t ∈ B J , called the space of initial conditions, consists of 2000 Mathematics Subject Classification. Primary 34M55; Secondary 32G08, 34M45.