Defining manifolds for Painlevé equations (Q2744101)

The \(J\)th Painlevé equation, \(J\)=\(I,II,\dots, VI\), is equivalent to a Hamiltonian system NEWLINE\[NEWLINE dx/dt=\partial H_J/\partial y, \qquad dy/dt=-\partial H_J/\partial x, NEWLINE\]NEWLINE where \(H_J=H_J(x,y,t)\) is a polynomial of \(x\) and \(y\) whose coefficients are rational functions of \(t\). All solutions \((x(t),y(t))\) define a foliation of a fiber space \((E_J,\pi_J, B_J)\) constructed by K.~Okamoto, which contains a trivial one \((\mathbb{C}^2\times B_J,\pi_J,B_J).\) The uniformity of the foliation is equivalent to the Painlevé property, namely the property that \(x(t)\) and \(y(t)\) are meromorphic in \(B_J.\) NEWLINENEWLINENEWLINEThe purpose of this article is to give a geometric interpretation of the original proof of the Painlevé property by P.~Painlevé and M.~Hukuhara through the treatment of the defining manifold \(E_J.\) The author discusses the Hamiltonian system corresponding to the \(IV\)th Painlevé equation, and the other cases are treated in a similar way.NEWLINENEWLINEFor the entire collection see [Zbl 0969.00055].