Recommend me a tube integrated amplifier

VAC makes very nice integrated units and used ones can be found near your price range. Used VAC equipment will also hold their value in the future. You will have no regrets.

You have a wonderful system and the perfect integrated would be the Audio Note Cobra. We have had one for awhile, replacing our Quad amp when it finally died. The Cobra runs our 45-year old Quad ESL-57s perfectly, and they are picky about type of power. The Cobra is very musical, always pulling you back to listen a bit more. 

I will throw my hat in the ring and suggest the OP look at the Ars-Sonum Filarmonia SM used unit ($2,950) I have listed for sale right now here on Audiogon. Push-Pull Class A pentode 28W/ch EL34 magic. Not fancy feature wise (no remote, no subwoofer output  but there is no problem with connecting a sub to the speaker terminals via speaker level input). Great build quality and since I am the USA dealer/distributor for these amps I can offer a 90 day warranty on the amp (30 days on tubes) and will even extend a full 30 day trial period with full refund on the amp.

Self bias operation means you just hook it up, turn it on, and it makes (great music).  Sound is not stereotypical warm/mushy tubes and not solid state like sounding Audio Research. Clean, pure, great tone and soundstage.

Cheers - Rich

+1 for Rogue's Cronus Magnum iii, but upgrade to the Dark edition at your price range. I have the Dark edition paired with Maggie 3.7s and a Perlisten D15s sub, and I love the combination. 

@erikt

Here is a fantastic Cary SLI-80 for sale.

https://www.facebook.com/share/9WXq2VHhPitW2y3m/?mibextid=kL3p88