Unitary monodromy implies the smoothness along the real axis for some Painlevé VI equation, I

Abstract: In this paper, we study the Painlev\'{e} VI equation with parameter $(\frac {9}{8},\frac{-1}{8},\frac{1}{8},\frac{3}{8})$. We prove (i) An explicit formula to count the number of poles of an algebraic solution with the monodromy group $D_{N}$, where $D_{N}$ is the dihedral group of order $2N$. (ii) There are only four solutions without poles in $\mathbb{C}\backslash \left { 0,1\right } $. (iii) If the monodromy group of the associated linear ODE of a solution $\lambda \left( t\right) $ is unitary, then $\lambda ( t) $ has no poles in $\mathbb{R}% \backslash { 0,1} $.