White to move and mate in two
The hints below could be of hep if you want to try to solve the Study; otherwise simply look at the solution published underneath the hints.
Black's King cannot move at all but they still have three pawns that can, and the f-pawn can even choose to move by one or two squares. All those potential moves will have to be taken into account in the search for our "key" first move.
If we give the move to Black, there are a few mates that immediately comes to mind: if ...f6 then Nf4#, if ...f5 Qd7# and if ...e4 Qxe4#. So for example a waiting move like 1. Ba2 is interesting but unfortunately insufficient because of 1. ...d6! and there is no immediate mate. But we are close: the problem is that ...d6 clears and escape square on d7 for the King, so we have to find a move that, among other things, protects d7 someway.
The key move for this problem is the surprising 1. Ba4!! It is definitely not a waiting move because it gives Black more moves than they had initially, but every move by Black leads now to a mate in 1. Let's check:
1. ... e4 2. Qxe4#
1. ... f6 2. Ndc7#
1. ... f5 2. Qg8#
1. ... Kxd5 2. Bb3#
1. ... d6 2. Nbc7#
Note the crucial protection of d7 by the Ba4 in the last variation.