Surfaces with $chi=5, K^{2}=9$ and a canonical involution

In this paper, we classify the minimal surfaces of general type with $chi=5$, $K^{2}=9$ whose canonical map is composed with an involution. We obtain 6 families, whose dimensions in the moduli space are 28, 27, 33, 32, 31 and 32 respectively. Among them, the family of surfaces having a genus 2 fibration forms an irreducible component of $mathfrak{M}_{chi=5, K^{2}=9}$.