  In the present note we show that any surface of general type with
$p_g=2$,$q=1$ and non-birational bicanonical map has a pencil of curves of
genus 2. Combining this result with previous ones, one obtains thatan irregular
surface $S$ of general type with $\chi(S)\geq 2$ and non-birational bicanonical
map has a pencil of curves of genus 2.
