I’ve been researching this a bit on the superspecialised web sites (majorgeeks, bleeping computer and what have you).
Apparently, the software protects itself from AV scans through changes it has made to the system at the registry level of even deeper.
One of the tools that is most frequently recommended for removing this protection in order to get at the malware is Kaspersky’s ‘TDSSkiller.exe’.
I can get this on to my machine, but it won’t run even when renamed (although it will on my other systems).

The standard procedure thereafter seems to involve the tools DDR, Gmer and Combofix and much expert analysis of the logs produced by these tools. The humble user is advised not to attempt to use these tools without supervision, so I have not.

Also, there are any number of support threads of this type which go on for pages (and weeks) and finish up with a still infected system.

The main symptoms I have are two bogus instances of iexplore.exe which start up shortly after bootup and without my having opened IE.
There seem to be a number of sites at the end of the command line that opens. If I close down one, the new process that opens to replace it is likely to have a different site in the command line, but they all appear to be equally suspect. These URLs are clearly written into the software (malware) as they appear when my connection is disabled.

The other symptom – which I appear to experience less frequently, if at all now – is URL redirect following searches with search engines.

When my internet connection is enabled, I note (from the BitDefender activity window) that data is being downloaded through these processes into my system; so far I have never seen any outbound data (i.e. data being uploaded from my system).

My guess now is that this infection pre-dated the installation of the ‘WindowsRepair’ virus, which must have been downloaded on to the PC through these processes. Indeed this kind of unwanted installation may be the very point of the iexplore.exe infection.

“Reformatting often results in a loss of a lot of data, and I try to avoid that first, even if time is a factor. Depends on how recent your backup was.”

Surely reformatting results in the loss of all data?
I can backup now (the system is still very much up and operational).
I have scanned my external drive (MalwareBytes, BitDefender) and it comes up clean, but then again so does my internal drive.

If there were just some failsafe way to stop any process called iexplore.exe from running, then that would help.

Thanks a lot Bob for your help