  We introduce the Schlesinger transformations of the Gambier equation. The
latter can be written, in both the continuous and discrete cases, as a system
of two coupled Riccati equations in cascade involving an integer parameter n.
In the continuous case the parameter appears explicitly in the equation while
in the discrete case it corresponds to the number of steps for singularity
confinement. Two Schlesinger transformations are obtained relating the
solutions for some value $n$ to that corresponding to either n+1 or n+2.

Exactly Solvable and Integrable Systems;Exactly Solvable and Integrable Systems