I have a constant water flow (it is a slow drip) into the reservoir of my Elektra T1 Sixties machine (see picture, inflow pipe on right side of picture). If I place my finger over the tube it stops so it isn't very high pressure, but it is incessant. Does anyone know what might be causing this problem and if it is easily fixed? This occurs both with the machine on and off. Any help much appreciated...

Assuming that the inline pressure is not an issue and the flow/dripping does NOT stop after the machine is only heating up initially (which is normal for the expansion valve to be doing)
most likely your exp. valve seat is worn out
you can see replacement on our site you you like.

I don't have any pressure regulation on the inline, the only thing between the valve and the espresso machine is a (resin/salt) water softener. Does the water pressure need to be regulated?

Max input pressure on the sixties series machines is 4 bar. If you are over that, you need a pressure regulator. I run my machine at 3 bar mains but that valve does not open until you get to a much higher pressure that that as I recall. So your valve seals are probably worn.

Great advice. I will probably both replace the expansion valve (I take it this is the valve leading into the reservoir) and install a pressure regulator.

Two more questions come up:
1) Where is the pressure regulator installed, before or after the water softener
2) When installing the new expansion valve, do the pipes move enough to allow removal of the old and installation of the new valve, or is there a certain technique that should be used for replacing?

I would get a professional to do the fix, but I am not aware of any Elektra repair places in my location (Luxembourg), so unless it is really tricky (in which case I would take the machine to Germany) I would like to try to do it myself (I have decent plumbing skills and have even replaced a water pump on an ECM Giotto)