Example 3 Find the solution to $907 x_{1}+731 x_{2}=2107$.

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Solving, if there is a solution, then 731 must divide $-907 x_{1}+2107$, so from the original equation we get
$$\begin{aligned}
x_{2}= & \frac{1}{731}\left(-907 x_{1}+2107\right) \\
= & -x_{1}+3+\frac{1}{731} \\
& \cdot\left(-176 x_{1}-86\right)
\end{aligned}$$

Thus, the original equation is equivalent to the indeterminate equation
$$x_{3}=\frac{1}{731}\left(-176 x_{1}-86\right) \in \mathbf{Z}^{\oplus}$$

Similarly, solving for
$$\begin{aligned}
x_{1}= & \frac{1}{176}\left(-731 x_{3}-86\right) \\
= & -4 x_{3} \\
& +\frac{1}{176}\left(-27 x_{3}-86\right)
\end{aligned}$$

The original indeterminate equation is equivalent to the indeterminate equation
$$x_{4}=\frac{1}{176}\left(-27 x_{3}-86\right) \in \boldsymbol{Z}$$

Similarly, solving for
$$\begin{aligned}
x_{3} & =\frac{1}{27}\left(-176 x_{4}-86\right) \\
& =-7 x_{4}-3+\frac{1}{27}\left(13 x_{4}-5\right)
\end{aligned}$$
(Continued on the left column of the next page) The original indeterminate equation is equivalent to the indeterminate equation
$$x_{5}=\frac{1}{27}\left(13 x_{4}-5\right) \in \boldsymbol{Z}$$ Similarly, solving for
$$\begin{aligned}
x_{4} & =\left(27 x_{5}+5\right) / 13 \\
& =2 x_{5}+\left(x_{5}+5\right) / 13
\end{aligned}$$

The original indeterminate equation is equivalent to the indeterminate equation
$$\begin{array}{l}
x_{6}=\left(x_{5}+5\right) / 13 \in \boldsymbol{Z} \\
x_{5}=13 x_{6}-5=-5+13 x_{6}
\end{array}$$

From this, we can trace back upwards, see the right column.
$$\begin{aligned}
x_{2}= & -x_{1}+3+x_{3} \\
= & -\left(-258+731 x_{6}\right)+3 \\
& +\left(62-176 x_{6}\right) \\
= & 323-907 x_{6}
\end{aligned}$$
$$\begin{aligned}
x_{1}= & -4 x_{3}+x_{4} \\
= & -4\left(62-176 x_{6}\right) \\
& +\left(-10+27 x_{6}\right) \\
= & -258+731 x_{6}
\end{aligned}$$
$$\begin{aligned}
x_{3}= & -7 x_{4}-3+x_{5} \\
= & -7\left(-10+27 x_{6}\right)-3 \\
& +\left(-5+13 x_{6}\right) \\
= & 62-176 x_{6}
\end{aligned}$$ $\downarrow$ (Continued from the right column of the next page)
† (Continued from the right column of the previous page)
$$\begin{aligned}
x_{4} & =2 x_{5}+x_{6} \\
& =2\left(-5+13 x_{6}\right)+x_{6} \\
& =-10+27 x_{6}
\end{aligned}$$
Thus, we have found all solutions:
$$x_{1}=-258+731 x_{6}, \quad x_{2}=323-907 x_{6}, \quad x_{6}=0, \pm 1, \pm 2, \cdots$$