Given a meromorphic linear differential system with an arbitrary single level r ≥ 1, we build a regular holomorphic perturbation which preserves the single level and we show that the Stokes-Ramis matrices of the initial system are limits of convenient products of the perturbed ones. As an application, we provide an alternative method for the effective calculation of the Stokes multipliers of the initial system illustrated on two examples. No assumption of genericity is made on the initial system.

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A functional equation Σ^m_i=1a_iu(φ_i(z)) = f(z) is considered, where {φ_i(z)^m_i=1 are holomorphic functions in a neighborhood of z = 0 with φ_i(0) = 0 and f(z) is holomorphic in a sector with vertex z = 0. It is shown under some conditions of {φ_i(z)^m_i=1, f(z) and {a_i}^m_i=1 that the equation has a formal power series solution that is Borel summable.

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We study the reduction problem for a holomorphic singular vector field with nilpotent linear part at the origin, P = (y + a(X)) ∂_x + (z + b(X)) ∂_y + c(X)∂_z, where X = (x, y, z) ∈ C³. By introducing a notion of quasi-valuation, which was given in the previous paper [M-S] with A. Shirai, we characterize a class which can be reduced into simpler ones by formal change of coordinates which admit various variations in Theorems A and B.

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For the Painlevé equations (I) through (V) near the point at infinity we present several families of two-parameter solutions. Our solutions are expressed by asymptotic series with coefficients polynomial in exponential terms, and also by convergent power series in exponential terms with coefficients expanded into asymptotic series. Both expressions are valid without a restriction on integration constants. We propose a direct method to derive asymptotic solutions, which is also applicable to more general nonlinear equations. As applications of our results, for general solutions of the third and the fifth Painlevé equations, we give estimates for the number of a-points including poles in given sectors.

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