Is there anything known about isolated conics in a del Pezzo surface: their number, arrangement, and the corresponding elements of the class group of surface's minimal desingularization? (Isolated means not belonging to a continuous family of conics in the surface.)

A description similar to the one for isolated lines would be of most interest: "A del Pezzo surface has only finitely many lines. They correspond to curves E such that E^2 = E·K = −1 (so-called -1-curves) on the desingularization."

More specifically, the question is about del Pezzo surfaces of degrees 5 and 6. References not requiring much background in algebraic geometry are greatly appreciated.

[Edit] And if we have a surface in C^3, whose linear normalization is a degree 5 or 6 Del Pezzo surface, can we say anything about isolated conics in this situation?

[Edit2] I have found the following related result in the literature:

"Any surface is a projection from its linear normalization. The projection is birational, and it preserves the degree of the surface and the degree of any curve not contained in the singular locus."

Notice that the conics contained in the singular locus are also interesting for me.

Additional question about surfaces in C^3 still unanswered.