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Abstract. We study special solutions of the fifth Painlevé equation which are analytic around t ¼ 0. We calculate in particular the linear monodromy of those solutions exactly. We also show how those solutions are related to classical solutions in the sense of Umemura.

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§ 0. IntroductionIn his lectures at Kyoto University, Professor M. Sato presented a program for generalizing the soliton theory ([9] ; cf. [10]). The KadomtsevPetviashvili (KP) equation is a typical example of the soliton theory. The KP equation is written in the form of deformation equations of a linear ordinary differential equation. The time evolutions of a solution are interpreted as dynamical motions on an infinite dimensional Grassmann manifold ([7], [9]). The Lie algebra of microdifferential operators of one variable acts on this manifold transitively. He conjectured that any integrable systems can be written in the form of deformation equations of a linear system, and proposed to investigate a deformation of differential equations in higher dimensions. He showed a simple example of a deformation of holonomic systems in higher dimensions ([9]), and its generalization is treated in [4]. In this paper we study a deformation of ^-modules in higher dimensions.First we review the KP equation. We denote by 8 the ring of microdifferential operators of one variable x. We fix a microdifferential operator P, and denote by t P a time variable with respect to P. We study the following evolution equation associated to P:where W=W(x, D^l+^^w^D^e. We denote by

We classify solutions of the third Painlevé equation P III , which are meromorphic around the origin, and determine their linear monodromy. Combined with [4], [5], all of meromorphic solutions of the Painlevé equations around fixed singular points of the Briot-Bouquet type are completely determined. We characterize the linear monodromy of meromorphic solutions for the third, fifth and sixth Painlevé equations.

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