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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 solutionwith 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\} $.

Unitary monodromy implies the smoothness along the real axis for some Painlev\'{e} VI equation, I