Hyperbolic Roussarie fields with degenerate quadratic part

In many problems in analysis and geometry there is a need to investigate vector fields with singular points that are not isolated but rather form a submanifold of the phase space, which most often has codimension 2. Of primary interest are the local orbital normal forms of such fields. ‘Orbital’ means that we may multiply vector fields by scalar functions with constant sign. In what follows, all vector fields and functions are assumed without mention to be smooth (of class C∞) unless otherwise stated. Roussarie [1] investigated vector fields of a certain special type which satisfy the following conditions at all singular points: 1) the components of the field lie in the ideal (of the space of smooth functions) generated by two of the components; 2) the divergence of the vector field (the trace of its linear part) is zero. We call such fields R-fields after Roussarie. In local coordinates the germ of an R-field has the following form at its singular point: