  In this paper, we study the Darboux equations in both classical and system
form, which give the elliptic Painlev\'e VI equations by the isomonodromy
deformation method. Then we establish the full correspondence between the
special Darboux equations and the special Painlev\'e VI equations. Instead of
the system form, we especially focus on the Darboux equation in a scalar form,
which is the generalization of the classical Lam\'{e} equation. We introduce a
new infinite series expansion (in terms of the compositions of hypergeometric
functions and Jacobi elliptic functions) %around each of the four regular
singular points of the for the solutions of the Darboux equations and regard
special solutions of the Darboux equations as those terminating series. The
Darboux equations characterized in this manner have an almost (but not
completely) full correspondence to the special types of the Painlev\'e VI
equations. Finally, we discuss the convergence of these infinite series
expansions.
