Let S be an algebraic surface in 3-dimensional complex projective space. Suppose that:
The degree of S is either 5 or 6;
The generic plane section of S is a curve of genus 1.
(Equivalently, the linear normalization of S is a Del Pezzo surface of degree 5 or 6).
Can S have singular conics (= degree 2 curves consisting of singular points)?
If the answer is positive then what are possible numbers of intersection points of a singular conic with a nonsingular conic contained in S?
This is a ``continuation'' of this question. References or answers not requiring much background in algebraic geometry are especially appreciated.