Let $a$ and $b$ be integers. Find all integer solutions that satisfy $(a, b) + 9[a, b] + 9(a + b) = 7ab$.

The text translated to English is as follows:

Let $a$ and $b$ be integers. Find all integer solutions that satisfy $(a, b) + 9[a, b] + 9(a + b) = 7ab$.

The problem is to find all pairs of integers $(a, b)$ that satisfy the equation $(a, b) + 9[a, b] + 9(a + b) = 7ab$, where $(a, b)$ denotes the greatest common divisor (gcd) of $a$ and $b$, and $[a, b]$ denotes the least common multiple (lcm) of $a$ and $b$.
Let $(a, b)=k$, then $a=k x, b=k y$, where $x, y \in N$, and $(x, y)=1$, then $[a, b]=k x y$. Substituting into the original equation, we get
$$
1+9 x y+9(x+y)=7 k x y .
$$

Solve the equation (1) to determine the range of values for the parameter $k$.
From (1) $7 k=\frac{1}{x y}+9\left(\frac{1}{x}+\frac{1}{y}\right)+9$.
$$
\because \frac{1}{x y}+9\left(\frac{1}{x}+\frac{1}{y}\right)+9=\left(\frac{1}{x}+9\right)\left(\frac{1}{y}+9\right)-72
$$
$\therefore(1+9)(1+9)-72=28$.
$\therefore 7 k \leqslant 28$, so $k \leqslant 4$.
For better narration, assume $x \leqslant y$.
(1) When $k=1$, the equation (1) becomes $1+2 x y+9(x+y)=0$, which has no integer solutions.
(2) When $k=2$, the equation becomes $5 x y-9(x+y)=1$.
1. Multiply both sides by 5 and factorize, we get
$$
\begin{array}{l}
(5 x-9)(5 y-9)=86=2 \times 43=1 \times 86 . \\
\therefore\left\{\begin{array} { l } 
{ 5 x - 9 = 2 , } \\
{ 5 y - 9 = 43 ; }
\end{array} \quad \left\{\begin{array}{l}
5 x-9=1, \\
5 y-9=86 .
\end{array}\right.\right.
\end{array}
$$

The former has no positive solutions, the latter has the solution $x=2, y=19$.
(3) When $k=3$, the equation (1) becomes $12 x y-9(x+y)=1$.
$\because$ the left side $\equiv 0(\bmod 3)$, the right side $\equiv 1(\bmod 3)$,
$\therefore$ the equation (1) has no positive integer solutions.
(4) When $k=4$, the equation (1) becomes $19 x y=9(x+y)+1$.
$\because 1 \leqslant x \leqslant y$,
$\therefore 19 x y=9(x+y)+1 \leqslant 9(y+y)+y=19 y$,
$\therefore x \leq 1$, hence $x=1$.
Solving, we get $y=1$.
$\therefore$ The solution to the equation (1) is $x=y=1$.
In summary, by the symmetry of $x, y$, the solutions to the equation (1) are
$$
\left\{\begin{array} { l } 
{ x = 2 , } \\
{ y = 1 9 ; }
\end{array} \left\{\begin{array} { l } 
{ x = 1 9 , } \\
{ y = 2 ; }
\end{array} \left\{\begin{array}{l}
x=1, \\
y=1 .
\end{array}\right.\right.\right.
$$