Defining Manifolds for Painlvé Equations

where HJ = HJ(x, y, t) is a polynomial of x and y of which the coefficients are rational functions of t holomorphic in BJ := P−ΞJ , ΞJ 3 ∞ being the fixed singular points of the J-th Painlevé equation (Ref. 2). Here the equivalece means that the second order nonlinear differential equation in x obtained from the J-th Painlevé system by elimination of y is just the J-th Painlevé equation. The J-th Painlevé system defines a nonsingular foliation of the trivial fiber space (C×BJ , πJ , BJ) every leaf of which is transversal to fibers. However, the foliation is not uniform. (Uniformity means that for any point P0 ∈ C × BJ , every curve in BJ with starting point πJ(P0) can be lifted on the leaf passing through the point P0.) In the paper Ref. 5, K. Okamoto constructed, for each J , a fiber space (EJ , πJ , BJ ) having the (C × BJ , πJ , BJ ) as a fiber subspace so that the extension of the J-th Painlevé system defines a uniform foliation of EJ . The uniformity of the foliation is equivalent to the so-called Painlevé property which is stated as: for any (x0, y0, t0) ∈ C × BJ , x(t) and y(t) can be meromorphically continued along any curve in BJ with starting point t0, where (x(t), y(t)) is the local solution of the J-th Painlevé system satisfying the initial condition (x(t0), y(t0)) = (x0, y0). Okamoto called each fiber EJ(t) := π−1 J (t) the space of initial conditions, because there is a bijection from EJ(t) to the set of all solutions of the J-th Painlevé system, however he did not name the total space EJ itself. So I call it defining manifold in this note. Okamoto constructed each fiber EJ (t) (t ∈ BJ) by compactification of C × t, 8 times quadratic transformations (the result is denoted by