We consider solutions of the word equation $X_1^2\cdots X_n^2={(X_1\cdots X_n)}^2$ such that the squares $X_i^2$ are minimal squares found in optimal squareful infinite words. We apply a method developed by the second author for studying word equations and prove that there are exactly two families of solutions: reversed standard words and words obtained from reversed standard words by a simple substitution scheme. A particular and remarkable consequence is that a word w is a standard word if and only if its reversal is a solution to the word equation and $\gcd (|w|,|w{|}_1)=1$. This result can be interpreted as a yet another characterization for standard Sturmian words.

We apply our results to the symbolic square root map $\sqrt{\cdot }$ studied by the first author and M. A. Whiteland. We prove that if the language of a minimal subshift Ω contains infinitely many solutions to the word equation, then either Ω is Sturmian and $\sqrt{\cdot }$-invariant or Ω is a so-called SL-subshift and not $\sqrt{\cdot }$-invariant. This result is progress towards proving the conjecture that a minimal and $\sqrt{\cdot }$-invariant subshift is necessarily Sturmian.

我们考虑方程的解 $X_{\mathrm{1个}}^2\cdots X_{ñ}^2={（X_{\mathrm{1个}}\cdots X_{ñ}）}^2$ 这样的正方形 $X_{\mathrm{一世}}^2$是在最佳平方无穷词中找到的最小平方。我们应用第二作者开发的一种方法来研究单词方程，并证明存在两种完全正确的解决方案系列：反向标准单词和通过简单替换方案从反向标准单词获得的单词。一个特殊且引人注目的结果是，当且仅当单词w的反转是单词方程的解并且$\mathrm{光盘}（|w|，|w{|}_{\mathrm{1个}}）=\mathrm{1个}$。这个结果可以解释为标准Sturmian单词的又一个表征。

我们将结果应用于符号平方根图 $\sqrt{\cdot }$由第一作者和MA Whiteland研究。我们证明，如果最小子移位Ω的语言包含对单词方程式的无限多个解，则Ω是Sturmian且$\sqrt{\cdot }$-不变或Ω是所谓的SL-subshift，而不是 $\sqrt{\cdot }$-不变。这个结果正在朝证明最小和最小$\sqrt{\cdot }$-不变亚变位必然是Sturmian。