  By solving the infinitesimal Galilean symmetry for the KdV hierarchy, we
obtain an explicit expression for the corresponding one-parameter Lie group,
which we call the Galilean symmetry of the KdV hierarchy. As an application, we
establish an explicit relationship between the non-abelian Born--Infeld
partition function and the generalized Br\'ezin--Gross--Witten partition
function.
