My favorites (not necessarily recommendations--precede with caution: I accept responsibility only for my own taste
) :
1 - Previn and Boult EMI (we'll soon see if the BBC Boult will displace that)
2 - Sinopoli and Tate, both about as different from Solti as can be imagined.
Sarge
Beautiful description! All of those works you mention are very special to me. Overall, I could easily say Elgar is one of the most special composers to me. A composer who I have grown up with, who I have enjoyed from the very beginning of my musical life! 
As I have been saying, perhaps try Boult's EMI set, or Elder's set with the Halle. Also, I would really really recommend you get Downes' performance with the BBC Philharmonic on Naxos. The Solti is excellent, so you made a good first choice!
I really wished that Bernstein had recorded the symphonies. He did such an amazing job with the Enigma Variations, his recording is my favourite performance of the piece.
Haha, no problem, Sarge, and thanks for the feedback

I read very good reviews about Boult's recording, which was also recommended me by Daniel before; I think I may choose this one, then. Anyway, Previn on EMI looks very interesting too.
Thank you, Daniel

Very well, besides Boult's set, and eventually Previn's one, I may also consider Downes' performance. Yes, absolutely, Bernstein's
Enigma Variations is simply outstanding, impressive; I wished that he had recorded the symphonies too, I'm sure they would have been excellent. But at least he made some performances of Elgar's pieces, Karajan never did it.

Certainly, I know Elgar's music is very important for you, that's wonderful it is able to strike and inspire you in a so deep way; I could easily share the feeling about me and Johann Strauss as I've grown up listening to the
Neujahrskonzerte.
