I don't think it's quite fair to categorically rule out all solid state amps as being good matches for the HD600/650's. For example, many users on this site (not just me) are of the opinion that the Nuforce HDP (DAC/Amp) has a particularly great synergy with the HD650's. I believe the engineers and designers of the HDP even used the HD650s as one of the headphones they were targeting to sound great with HDP. In any event, imo, the sound of the HDP with the HD650's is very lush, warm, smooth, transparent, spacious and natural despite the fact that it's a SS and not a tube amp. Also, imo, the tendency of the HDP to sound somewhat bright and detailed seems to mix quite well and act as somewhat of a counterbalance to the lush warm/dark HD650s. All imo.
I am not trying to say that SS amps is a bad match. However I am trying to state that the ST and M-stage will be a very poor match, from my own experience - which could mean that a lot of amps in the same league will be as well.
There is very much likely a SS amp which will synergize perfectly well, which is what i meant when i wrote that the mids should be in focus and not the highs/lows to remove the veil.
Also the last line I wrote was key; I merely suggested to try a tube amp.
However reading my post I can see how it can be interpretted as very anti-SS.