We first rearrange so that the same variables are on the same side: . Notice that is a perfect square and if so is, then we can factor the LHS, this motivates me to discuss the parities of .

If is odd, then when , . When , if , then and which is impossible since is odd. If , there's no solution. Hence is the only solution when is odd.

If is even, let , where . Our equation is now: . We now take the usual analysis: Let . From (1)+(2): , since is not divisible on the LHS, it must be true that (if , then (2)>(1) which is impossible). Hence .
Now the value of depends on the largest power of on the RHS when . On the other hand, (2)-(1) gives (this implies , ). If or , we can deduce that there's no solution: ${2^{a-3}3^n=a-1+2^{n-a-3}$ or , note that .

Now it just leaves the case when or . This means . Since is divisible by but not any larger powers, therefore is the largest power that divides the RHS, therefore where is the only positive solution, this solution is shown above that it leads to no solution. Hence in conclusion is our only solution.