Smooth local normal forms of hyperbolic Roussarie vector fields

In 1975, Roussarie studied a special class of vector fields, whose singular points fill a submanifold of codimension two and the ratio between two non-zero eigenvalues $\lambda_1:\lambda_2=1:-1$. He established a smooth orbital normal form for such fields at points where $\lambda_{1,2}$ are real and the quadratic part of the field satisfied a certain genericity condition. In this paper, we establish smooth orbital normal forms for such fields at points where this condition fails. Moreover, we prove similar results for vector fields, whose singular points fill a submanifold of codimension two and the ratio between two non-zero eigenvalues $\lambda_1:\lambda_2=p:-q$ with arbitrary integers $p,q \ge 1$.