Genus $2$ pencils on surfaces with $p_g=K^2=1$, envelopes, and conics tangent to plane cubic curves

We consider $(1,1)$-surfaces, namely, minimal compact complex surfaces $S$ with $p_g (S) =K_S^2=1$: for these the bicanonical map is a covering of degree $4$ of the plane $P^2$. And we answer a question posed by Meng Chen, whether they can contain a genus 2 pencil (this is the standard reason of f