  Consider the space $C$ of conjugacy classes of a unitary group $U(n+m)$ with
respect to a smaller unitary group $U(m)$. It is known that for any element of
the space $C$ we can assign canonically a matrix-valued rational function on
the Riemann sphere (a Livshits characteristic function). In the paper we write
an explicit expression for the natural measure on $C$ obtained as the
pushforward of the Haar measure of the group $U(n+m)$ in the terms of
characteristic functions.
