ZX Spectrum Z80 Keeps Resetting

For some time now I have had a couple of ZX Spectrum machines. Neither works, each has a different problem. Today I got one of them out for another look. The problem with it is that it seems to constantly reset itself, you see the copyright screen briefly and then it disappears (the TV screen shows the usual "snow" when there is no signal). It just cycles round doing this. I put a logic analyser on it as well as a scope. The CPU Reset pin looks fine, it goes high and stays high. The logic analyser shows that it is happily reading code from the ROM during the boot sequence and it will suddenly start fetching instructions from address 0 again, this appears to be somewhat random. I don't have a lot of experience with logic analysers, and I have found that some pod/probe combinations cause the machine not to work at all, so the logic analyser is not, unfortunately, above suspicion. However the behaviour I see appears consistent with what the logic analyser is telling me. My thoughts are that this must either be a bad Z80 CPU or a bad ROM. Neither is socketed and I am reluctant to desolder ICs unless I really have to as there is always a risk of damage to the board. Has anyone seen a similar problem before? Could this indeed be the CPU or the ROM, or could there be a different cause?