Best Integrated Amp (for me)

Rega IO. I just picked up this lovely little integrated up @ $599.00 new, delivered. All I can say is wow! On board phono preamp, with 2 additional inputs. The phono is very nice indeed. With a newly purchased new Rega DAC-R (inbound streamed digital from MacBook Air) and Linn LP12 turntable outbound to ProAC speakers, it’s an all British system. Rega IO.
Edit: As others have referenced, the Rega Brio-R would be a great choice as well. 😎. More power than Rega IO, it has the all important phono preamp built in (as with the IO) and 2 more additional inputs if you should need them. Got Rega?

Out of the box- Musical Fidelity a3.2 integrated amp.  110wpc with an excellent phono stage.  Dual mono construction, and excellent sound.  Built to last another 20yrs.

Marantz has several very well regarded integrated amps that would be within your price range.

I currently have the Acurus DIA100 mkI partnered with Monitor Audio R700MD’s and I completely disagree with those above who said the mkII is bright.  The mkI, or at least my mkI, is in no way bright sounding at all, (although, I have read that sounding bright is the knock on some of these DIA’s).  
Mine is very beefy, sweet, and transparent sounding and I may be looking to sell it as I am moving overseas in a few months.  I am considering a smaller shoe-boxed size integrated.

I recently bought a Willsenton R8 integrated from China after it received very solid reviews. All hand wired KT88 integrated. I’m very happy. It’s my second KT88 integrated. I’m acquiring a collection. Great $1000 investment. 
https://m.youtube.com/watch?v=U0BDFT4f55w

As the owner of two pair of LS 50s, I can tell you that they definitely benefit from serious amplification. The Rogue Sphinx suggestion seems like a good one to me.