O. At a point p'~ pr-l(H), the stalks are (~-~q)p' = (~'-~q)p' = (~o-~q)p ' = (~q)pr(p') and the results follows from the holomorphic Poincar$'s lemma. Xn,,=O}. Let S be an open polysector at p in U.

We denote by ~'(c) For any two open subsets c, c' of T n, the restriction mapping ic, c of ~'(c') into ~'(c) is defined if ccc'. to see that {~'(c), We denote by ~ ' the It is easy icc,} becomes a presheaf which satisfies the sheaf conditions. the associated sheaf, and call it the sheaf of germs of functions strongly asymptotically developable to ~H' over T n . In the same manner, from the set ~o(C,r) = {fe~'(c,r); FA(f) = O} , we obtain the sheaf of germs of functions strongly asymptotically developable to 0 over T n.

As we recently noticed, a work of Sibuya in 1968 [70] also suggested research in this direction. In the following sections, in the notation of Part I, we prove existence theorems of integrable systems of partial differential equations of the first order under certain general conditions by developing Hukuhara's method [28]. In order to construct formal power series solutions of systems of differential equations, we provide analogues for systems of algebraic equations. ,x n as parameters. Xn,, e i ~ u = ai(x,u) i=l .....