In this paper we make essential steps in proving the finite
cyclicity of degenerate graphics in quadratic systems, having a line
of singular points in the finite plane. In particular we consider
the graphics $(DF_{1 a})$, $(DF_{2 a})$ of the program of [8]
to prove the finiteness part of Hilbert's 16th problem for quadratic
vector fields. We make a complete treatment except for one very
specific problem that we clearly identify.