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.