Instanton bundles on P1×F1

In this paper we deal with a particular class of rank two vector bundles (\emph{instanton} bundles) on the Fano threefold index one $F:=\mathbb{F}_1 \times \mathbb{P}^1$. We show that every instanton bundle $F$ can be described as cohomology monad whose terms are free sheaves. Furthermore prove existence for any admissible second Chern and construct nice component moduli space where they sit. Finally minimal (i.e. least possible degree class) aCM describe their space.