Gunther Reißig and Holger Boche.
A Normal Form for Implicit Differential Equations Near Singular Points.
Proc. 1997 Europ. Conf. on Circuit Th. and Design (ECCTD), Budapest, Hu., Aug. 30-Sept. 3, 1997, vol. 2, 1048-1053.

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Full text. (Free access.)

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Abstract:
We consider differential equations A(x)x'=g(x), where A is a (n x n)-matrix of C¹-functions and g is C¹. We investigate the above differential equation about singular points x₀ that are standard in the sense of Rabier. In particular, the null space of A(x₀) is of dimension 1. We show that there is a C¹-diffeomorphism that transforms the above equation about x₀ into
x₁ x₁' = +/-1,
x₂' = ... = x_n' = 0
about 0.

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