  We pursue the study started in \cite{Dem-Hmi} of the dynamics of the spectral
distribution of the free Jacobi process associated with one orthogonal
projection. More precisely, we use Lagrange inversion formula in order to
compute the Taylor coefficients of the local inverse around $z=0$ of the flow
determined in \cite{Dem-Hmi}. When the rank of the projection equals $1/2$, the
obtained sequence reduces to the moment sequence of the free unitary Brownian
motion. For general ranks in $(0,1)$, we derive a contour integral
representation for the first derivative of the Taylor series which is a major
step toward the analytic extension of the flow in the open unit disc.
