Hi Mike,
You mean you can get a 1/(A*s^2+1) type response without a gain? Not that we might actually use that as is, but I'll take another look at this. It seems to me that there would be an 's' term in the denominator no matter what choice of R's and C's we had.
LATER:
It seems that with the gain G at the output we can get other interesting responses. Maybe not as practical but worth looking into. For example, with a gain G=1 we are limited to whatever we can get with the R's and C's (that's the way it stands in the drawings), which means the response will be limited. However, with a gain G of:
G=(C4*R3)/(C3*R4)+C4/C3+1
we can actually get an infinite response. This of course is not practical as is, but may mean other interesting responses could be had. With both R's equal and both C's equal we can get an infinite response with a gain of G=3.
We could investigate this further.