We can se that $\overline{ \cdots 2}- \overline{ \cdots 4}= \overline{ \cdots 8}$ and $\overline{ \cdots 3}- \overline{ \cdots 6}= \overline{ \cdots 7}$ so $a \ne 2,3$. This means $a=1$ or $a=1$. Let $\underbrace{11\dots 11}_{n}=x$ then $\underbrace{11\dots 11}_{2n}-\underbrace{22\dots 22}_{n}= x \cdot (10^n-1)=9x^2$ and $\underbrace{44\dots 44}_{2n}-\underbrace{88\dots 88}_{n}= 4x \cdot (10^n-1)=36x^2$ which are perfect squares. Thus, $a=1$ or $a=4$.