2. Given that the length, width, and height of a rectangular prism are all integers, and the volume equals the surface area. Then the maximum value of its volume is


2.882.

Lemma When the sum of the reciprocals of two integers is a constant, the larger their difference, the greater their product.
Let the length, width, and height be $a$, $b$, and $c$, respectively, then
$$
a b c=2(b c+c a+a b) \Leftrightarrow \frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{1}{2} .
$$

Assume without loss of generality that $a \geqslant b \geqslant c$, then $c \geqslant 3$, and when $c=3$, we have $\frac{1}{a}+\frac{1}{b}=\frac{1}{6} \Rightarrow b \geqslant 7$.
When $b=7$, there is a solution $(a, b, c)=(42,7,3)$.
Therefore, the maximum value of $a b c$ is 882.
Note: All solutions under $a \geqslant b \geqslant c$ can be found to be $(42,7,3)$, $(24,8,3)$, $(18,9,3)$, $(15,10,3)$, $(12,12,3)$, $(20,5,4)$, $(12,6,4)$, $(8,8,4)$, $(10,5,5)$, and $(6,6,6)$, but it is more complicated.