  We classify the minimal surfaces of general type with $K^2 \leq 4\chi-8$
whose canonical map is composed with a pencil, up to a finite number of
families.
  More precisely we prove that there is exactly one irreducible family for each
value of $\chi \gg 0$, $4\chi-10 \leq K^2 \leq 4\chi-8$. All these surfaces are
complete intersections in a toric $4-$fold and bidouble covers of Hirzebruch
surfaces. The surfaces with $K^2=4\chi-8$ were previously constructed by
Catanese as bidouble covers of $\PP^1 \times \PP^1$.
