ADMISSIBLE SOLUTIONS OF THE SCHWARZIAN DIFFERENTIAL-EQUATION

Let R(z, w) be a rational function of w with meromorphic coefficients. It is shown that if the Schwarzian equation (*) {w, z}m = R(z, w) possesses an admissible solution, then d + 2m SIGMA-l(j=1)delta(alpha-j, w) less-than-or-equal-to 4m, where alpha-j are distinct complex constants. In particular, when R(z, w) is independent of z , it is shown that if (*) possesses an admissible solution w(z) , then by some Mobius transformation u = (aw + b)/(cw + d) (ad - bc not-equal 0), the equation can be reduced to one of the following forms: [GRAPHICS] where tau-j (j = 1,...,4) are distinct constants, and sigma-j (j = 1,...,4) are constants, not necessarily distinct.