  The Brezin-Gross-Witten tau function is a tau function of the KdV hierarchy
which arises in the weak coupling phase of the Brezin-Gross-Witten model. It
falls within the family of generalized Kontsevich matrix integrals, and its
algebro--geometric interpretation has been unveiled in recent works of Norbury.
We prove that a suitably generalized Brezin-Gross-Witten tau function is the
isomonodromic tau function of a $2\times 2$ isomonodromic system and
consequently present a study of this tau function purely by means of this
isomonodromic interpretation. Within this approach we derive effective
formul\ae\ for the generating functions of the correlators in terms of simple
generating series, the Virasoro constraints, and discuss the relation with the
Painlev\'{e} XXXIV hierarchy.
