On singularities of autonomous implicit ordinary differential equations

Implicit ordinary differential equations A(x)(x) over dot = g(x) with A(x) being an n x n matrix depending on x are investigated about their singular points, i.e., inconsistent points in the closure of the set of consistent points. Sufficient conditions are given that ensure that around such points, the above differential equation can be diffeomorphically transformed into the normal form x(1)(r) . (x) over dot(1) = +/-1, (x) over dot(2) = 0, ..., (x) over dot(n-m) = 0, Xn-m+1 = 0,... , x(n) = 0 for some r greater than or equal to 1 and some m, 0 less than or equal to m < n. These results are applied to examples from several fields of application, including an electrical network having impasse points.