Example 4 Find all integer solutions of the indefinite equation $3 x^{2}-4 x y+3 y^{2}=35$.

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Solve: Multiply both sides of the equation by 3, and complete the square to get
$$(3 x-2 y)^{2}+5 y^{2}=105$$

From equation (1), we get
$$5 y^{2} \leqslant 105$$

Therefore,
$$|y| \leqslant 4$$

When $|y|=4$, $|3 x-2 y|=5$, at this time the solutions to the original equation are
$$(x, y)=(1,4),(-1,-4)$$

When $|y|=1$, $|3 x-2 y|=10$, at this time the solutions to the original equation are
$$(x, y)=(4,1),(-4,-1)$$

When $|y|=0,2,3$, $(3 x-2 y)^{2}$ are $105,85,60$ respectively. In these cases, the resulting equations clearly have no integer solutions.

The above discussion shows that the original equation has 4 solutions:
$$(x, y)=(4,1),(1,4),(-4,-1),(-1,-4)$$