2. Cut an $8 \times 8$ chessboard into $p$ rectangles without cutting any square, and this cutting method must satisfy (1) each rectangle has an equal number of black and white squares; (2) let $a_{i}$ be the number of white squares in the $i$-th rectangle, then $a_{1}<a_{2}<\cdots<a_{p}$. Find the maximum value of $P$ for which the above cutting method exists, and for this $P$, find all possible $a_{1}, a_{2}, \cdots, a_{p}$.
2. $\sum_{i=1}^{P} a_{i} \geqslant \sum_{i=1}^{P} i=\frac{P(P+1)}{2}, \frac{P(P+1)}{2} \leqslant 32, P \leqslant 7, \sum_{i=1}^{7} a_{i}=32$, there are the following
5 solutions:
$$
\begin{array}{c}
1+2+3+4+5+6+11 \\
1+2+3+4+5+7+10 \\
1+2+3+4+5+8+9 \\
1+2+3+4+6+7+9 \\
1+2+3+5+6+7+8
\end{array}
$$

Among them, $1+2+3+4+5+6+11$ is not possible, because $a_{i} \neq 11$, the other four groups can all construct solutions.