One of the main goals of this
paper is to develop an algorithm for reducing the first order (singular) system of
differential equations:
to a Turrittin-Levelt canonical form. Here A(z) = zrAr+ zr+1Ar+1+⋯, r < −1 and
Ar+m∈gl(n;C) m ≥ 0. The reduction of (†) to a canonical form is implemented by
the natural gauge adjoint action of GL(n;ℱ) where ℱ is the algebraic closure
of the field of formal Laurent series about 0 with at most a finite pole at
0. For example, it is shown that the irregular part of the canonical form
(†) is determined by Ar+m,0 ≤ m < n(|r|− 1). The proofs utilize group
theoretic techniques as well as the method of Galois descent. In particular
almost all of the results generalize to the case where GL(n) and gl(n) are
replaced by an arbitrary affine algebraic group G over C and its Lie algebra
g.
