On singularities of autonomous implicit ordinary differential equations

Implicit ordinary differential equations A(x)x/spl dot/=g(x) with A(x) being an n/spl times/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/sub 1//sup r//spl middot/x/spl dot//sub 1/=/spl plusmn/1, x/spl dot//sub 2/=0,...x/spl dot//sub n-m/=0, x/sub n-m+1/=0,..., x/sub n/=0 for some r/spl ges/1 and some m, 0/spl les/m