On algebraic solutions of the fifth Painlevé equation

Abstract

According to [1, p. 102], it suffices to consider the fifth Painlev´e equation w′′ = 3w − 1 2w(w − 1) w′2 − w′z + (w − 1)2 z (αw + βw) + γw z + δw(w + 1) w − 1 , (P5) where α, β, γ, and δ are arbitrary complex parameters, in two cases (neglecting scale transformations of w and z) : δ = 0, γ 6 = 0 and δ = −1/2. (The integrable case with γ = δ = 0 is excluded here.) If δ = 0 and γ 6 = 0, then Eq. (P5) can be reduced to the third Painlev´e equation.