Jonathan Welton -- Damn - Guenter Stertenbrink beat me to it!
There seem to be a few solutions, all hard to find. Here's one that has the benefit of bilateral symmetry:
..........
.......x..
..x..x....
...x......
........x.
..x...x...
.....x....
.x.....x..
....x.....
..........
(The diagonal symmety doesn't show up too well in ascii.) Is there a solution with four fold symmetry? I doubt it, but it could be. I am a recent (1 week old) discoverer of your site, and love it. QoRS was one of the first puzzles on your back pages that I investigated deeper, it being my sort of puzzle. I notice its inventor, Scott Purdy, has another puzzle which appeals to me, that of removing edges from Kn to form a unique Hamiltonian path. Looks interesting. I have a number of my own puzzles invented over the years which may be of interest to you, and which I'll post, given time in my busy schedule.