On singularities of autonomous implicit ordinary differential equations

Implicit ordinary differential equations A(x)x˙=g(x) with A(x) being an n×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 x₁^r·x˙₁=±1, x˙₂=0,...x˙_n-m=0, x_n-m+1=0,..., x_n=0 for some r≥1 and some m, 0≤m\n\n\t\t