First tube integrated amp

I use PSB synchrony One speakers and listen to classical music.

You might want check out Grant Fidelity. They have some Shuguang Treasure amps that look nice.

I decided to go with separates, but I checked with some professional reviewers and all like the Primaluna Prologue 2. I think they are on sale at Upscale Audio. With your budget, you could also get separates.

PMB

I use and really like Audio Space amplifiers. The AS-3i EL34 integrated lists at about $1400. There is also a larger integrated built around the KT88 tube for more money. And if you want a remote, the Galaxy 34 is a nice amp.

I have a PrimaLuna and like it very much. The Cary amps are good as well. I would not recommend the Rogue amps if you know you like the sound of the PrimaLuna's, the Rogue amps are quite a bit brighter - they are much higher powered in general.

If I was considering an integrated amplifier I would look at the Audio Research VSi60. It's about as good as it gets. If you are patient you will find one used in your price range.