On surfaces with $p_g=2$, $q=1$ and non-birational bicanonical map
  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. 
Ciro Ciliberto and Margarida Mendes Lopes