Let \((A,B)\) be a pair of partial isometries with domains \(\mathcal D_A\), \(\mathcal D_B\) and ranges \(\mathcal R_A\), \(\mathcal R_B\), respectively, closed subspaces of a Hilbert space \(\mathcal H\). A commuting unitary extension of \((A,B)\) is a pair \((\widetilde{A},\widetilde{B})\) of commuting unitary operators \(\widetilde{A}\) and \(\widetilde{B}\) on a Hilbert space \(\mathcal F \), which contains \(\mathcal H\) as a closed subspace, that extends \(A\) and \(B\), respectively. The authors consider the description of the set of all minimal commuting unitary extensions of such a pair \((A,B)\) of operators, that \(\mathcal D_B \subseteq \mathcal D_{A^k}\), \(\mathcal R_B \subseteq \mathcal R_{A^k}\) for all \(k\geq 0 \) and which commute in the following weak sense (see \textit{R. Bruzual} and \textit{M.Domínguez} [Acta Sci. Math. 66, No. 3--4, 623--631 (2000; Zbl 0980.43003)]): \(\langle A^{-k}B f,B g\rangle_{\mathcal H } = \langle f,A^{k}\rangle_{\mathcal H}\) for all \( f,g \in \mathcal D_B\) and \(k \geq 0\); the description is achieved via the Arov-Grossman methods (see \textit{D. Z. Arov} and \textit{L. Z. Grossman} [Sov. Math., Dokl. 27, 518--522 (1983); translation from Dokl. Akad. Nauk SSSR 270, 17--20 (1983; Zbl 0543.47010); Math. Nachr. 157, 105--123 (1992; Zbl 0777.47007)]). If \((S_{m,n})_{(m,n)\geq (0,0)}\) is a multiplicative family of partial isometries on \(\mathbb{Z}^2\) with the lexicographic order and \(A= S_{(1,0)}, B=S_{(0,1)}\), then there exists a pair of commuting unitary extension \((\widetilde{A},\widetilde{B})\) of \((A,B)\). A sufficient condition for \((\widetilde{A}^{n},\widetilde{B}^{m})_{(n,m)\in \mathbb{Z}^{2}}\) to be a unitary extension of the given family is presented and under that condition the authors give a description of the set of extensions.