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In this paper, we describe the minimal reducing subspaces of Toeplitz operators induced by non-analytic monomials on the weighted Bergman spaces and Dirichlet spaces over the unit ball B2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathbb{B}_2}$$\end{document}. It is proved that each minimal reducing subspace M is finite dimensional, and if dim M ≥ 3, then M is induced by a monomial. Furthermore, the structure of commutant algebra ν(Tw¯NzN):={MwN∗MzN,MzN∗MwN}′\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\nu ({T_{\overline w {N_z}N}}): = {\{ M_{{w^N}}^ * {M_{{z^N}}},M_{{z^N}}^ * {M_{{w^N}}}\} ^\prime }$$\end{document} is determined by N and the two dimensional minimal reducing subspaces of Tw¯NzN\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${T_{\overline w {N_z}N}}$$\end{document}. We also give some interesting examples. [ABSTRACT FROM AUTHOR]