Solving from the original equation, we get
$$\begin{array}{l} x_{2}=\frac{1}{21}\left(-117 x_{1}+38\right)=-6 x_{1}+2+\frac{1}{21}\left(9 x_{1}-4\right) \\ x_{3}=\frac{1}{21}\left(9 x_{1}-4\right) \in \boldsymbol{Z} \\ x_{1}=\frac{1}{9}\left(21 x_{3}+4\right)=2 x_{3}+\frac{1}{9}\left(3 x_{3}+4\right) \\ x_{4}=\frac{1}{9}\left(3 x_{3}+4\right) \in \boldsymbol{Z} \\ x_{3}=\frac{1}{3}\left(9 x_{4}-4\right)=3 x_{4}-1-\frac{1}{3} \end{array}$$

The last equation indicates: $x_{3}, x_{4}$ cannot both be integers, so the indeterminate equation has no solution.