  This paper is a companion to a series of papers devoted to the study of the
spectral distribution of the free Jacobi process associated with a single
projection. Actually, we notice that the flow solves a radial L\"owner equation
and as such, the general theory of L\"owner equations implies that it is
univalent in some connected region in the open unit disc. We also prove that
its inverse defines the Aleksandrov-Clark measure at $z=1$ of some Herglotz
function which is absolutely-continuous with an essentially bounded density. As
a by-product, we deduce that $z=1$ belongs only to the discrete spectrum of the
unitary operator whose spectral dynamics are governed by the flow. Moreover, we
use a previous result due to the first author in order to derive an explicit,
yet complicated, expression of the moments of both the unitary and the free
Jacobi processes. The paper is closed with some remarks on the boundary
behavior of the flow's inverse.
