Example 6 Find all integer solutions of the following indeterminate equation:
$$4 x-9 y+5 z=8$$
Let $t$ be an integer and have
$$4 x-9 y=t$$

From (29) and (30), we have
$$t+5 z=8$$

By $x=-2 t, y=-t$ being a set of integer solutions to (30), and by Theorem 1, we get that $x=-2 t+9 u, y=-t+4 u, u=0, \pm 1, \pm 2, \cdots$ are all integer solutions to (30). By Theorem 1 and $t=3, z=1$ being a set of integer solutions to (31), we get
$$t=3-5 v, \quad z=1+v$$

are all integer solutions to (31), where $v=0, \pm 1, \pm 2, \cdots$. Substituting $t=3-5 v$ into $x=-2 t+9 u$, we get
$$x=-6+10 v+9 u$$

Substituting $t=3-5 v$ into $y=-t+4 u$, we get
$$y=-3+5 v+4 u$$

From (32) to (34), we know that all integer solutions to (29) are
$$\begin{array}{l}
x=-6+10 v+9 u \\
y=-3+5 v+4 u \\
z=1+v
\end{array}$$

where $\boldsymbol{u}$ and $v$ are arbitrary integers.