PageHeader : Funkcialaj Ekvacioj, 40 (1997) 271-291 Title : On Some Hamiltonian Structures of Painleve Systems, I AuthorInfo : By AuthorInfo : Tsutomu SHIODA and Kyoichi TAKANO AuthorInfo : (Kobe University, Japan) section : 0. Introduction section : 1. Statement of main results section : 2. Review of the construction of the total space $E$ subsection : 2.1. The space $\overline{\Sigma_{\epsilon}}$ . subsection : 2.2. The first quadric transformations with centers $a_{v}^{(0)}(t)$ , $t\in B$ , $v=0,1$ , $t$ , $\infty$ . subsection : 2.3. The second quadric transformations with centers $a_{v}^{(1)}(t)$ , $t\in B$ , $v=0,1$ , subsection : 2.4. The space E. section : 3. Proof of Theorem 1 section : 4. Proof of Theorem 3 subsection : 4.1. Reduction of K to a polynomial. subsection : 4.2. Conditions on the coefficients of K. subsection : 4.3. Reduction of K to a polynomial of small degree. subsection : 4.4. Completion of the proof of Theorem 3. section : References