The dimension of the space of Garnier equations with fixed locus of apparent singularities

We show that the conditions imposed on a second order linear differential equation with rational coefficients on the complex line by requiring it to have regular singularities with fixed eigenvalues of the residue at the points of a finite set $P$ and apparent singularities at a finite set $Q$ (disjoint from $P$) determine a linear system of maximal rank. We conclude that the family of such differential equations is of the expected dimension.