The structure of twisted power partial isometries

Let $n>1$ and let $\{U_{ij}\}_{1\leq i<j\leq n}$ be $n\choose 2$ commuting unitaries on a Hilbert space $\mathcal{H}$. Suppose $U_{ji}:=U^*_{ij}$, $1\leq i<j\leq n$. An n-tuple of power partial isometries $(V_1,...,V_n)$ on Hilbert space $\mathcal{H}$ is called $\mathcal{U}_n$-twisted power partial isometry with respect to $\{U_{ij}\}_{i<j}$ (or simply $\mathcal{U}_n$-twisted power partial isometry if $\{U_{ij}\}_{i<j}$ is clear from the context) if $V_i^*V_j=U_{ij}V_jV^*_i, ~~ V_iV_j=U_{ji}V_jV_i ~~\text{and}~~ V_kU_{ij}=U_{ij}V_k~~(i,j,k=1,2,...,n,~\text{and}~i\neq j).$ We prove that each $\mathcal{U}_n$-twisted power partial isometry admits a Halmos and Wallen \cite{HW70} type orthogonal decomposition.