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\} $.
Categories: math.CA