Best Integrated Amp (for me)

All good choices above.
Personally, I'd pick up a vintage Marantz or Luxman from the '70s.
I own several that I use as described in your opening post, I've not had any issues.

A little patience will be required if one isn't listed but look for a used, Rega Elex R.  That speaker likes a little power are needs about 70w of good quality current to sound great.  The Brio is a little shy of what you need in terms of power but will match brilliantly with the Kefs.  

Plenty of good recommendations.  If you have a MM cartridge, consider the Heed Elixir.  Over budget, but you have good speakers and I would advise you not to cheap out too much or you will hear it.  My advice is to buy new so you have dealer support and warranty and spend a bit more to get something you will like a few years down the road. 

Rotel A series, A-11, A-12 and A-14 are all nice afforable Integrated Amps. If you need digital inputs, the 12 and the 14 have 2ea of opt and coax. I believe the 11 has all rca inputs.

The Lyngdorf TDAI 1120 is worth adding to your list. It has a phono input, analogue input, plus Internet radio and streaming AND Room Correction! It sounds fantastic for the money and will optimise the speakers for your room - no brainer 👍

I'm going to go ahead and suggest what I bought 2 years ago and it is still giving me great sound.

The Vincent SV-500 Hybrid Integrated. 

Simple looks in Black or Silver.

Slightly above your suggested budget but worth it in my opinion. 

The Parasound NewClassic 200 Integrated is really nice. 
I got one 2 months ago and have now sold off my preamp and power amp....no regrets.
Its very musical, outputs a clean 110 wpc @ 8 ohms, mm/mc phonostage, bass management, a good burr-brown DAC and a back-lit remote that gives you full control over bass, treble, balance, sub-gain, even a sub on/off. I do a lot of low-level listening so the tone controls can come in handy. 
Parasound has been experiencing component shortages the last few months making this integrated amp almost impossible to find. It's supposed to be back in stock by mid-late Summer. In the mean time the one for $900 that Bkeske mentioned is a great deal.

op

my suggestion of the hegel is solely for the quality of its amp section ... consider the dac and streaming features freebies

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@audioconnection

op said -- "I’m looking to spend somewhere between $500-$1000."

johnny, if you have a used belles aria available for under a grand i would like to buy it :)

Rogue Audio sphinx is a great option for you as well. New they're like 1600 but there's at least one for sale on here.

https://www.audiogon.com/listings/lisab5i4-rogue-audio-sphnix-v2-hybrid-integrated-amplifier-black-w...

https://www.marantz.com/en-us/product/amplifiers/pm8006

I had a Marantz PM8005 that I really enjoyed for several years before upgrading to tube. The current model is the PM8006 at $1100.  I would check to see if you can find a used PM8005 or PM8006 somewhere.  I sold my PM8005 for $650.

Arcam SA10 has the digital and phono hookup you need.About $800 new. It's predecessor, the FMJ-A19 was often used as a source for the LS 50's by What HIFI. Those can be had on the used market for $400 to $600. Great phono section, but no digital.

This REGA should fit the bill!
https://www.musicdirect.com/integrated-amp/Rega-Brio-2017-Integrated-Amplifier?utm_source=bing&u...

I used to have the Cambridge audio CXA60, which was a really nice looking and decent sounding amp with a good DAC on board. 

Then I upgraded to separates (Odyssey Khartago extreme monoblocks) and a tube pre and the difference was significant. I really like the Odyssey amps and would recommend looking into the Tempest integrated that sells for about 1100 I think.

Well, if you are open to vintage, I would look at Sansui, Kenwood, Marantz, Pioneer, Yamaha, Luxman, Accuphase, Harmon-Kardon, etc.

But, really, companies like Yamaha and Marantz (and many others) also make current integrated amp models that you may like, and are in your budget range. I’m a separates guy, so don’t keep up on the newest integrated models in that price range, and don’t know them off the top off my head. These all have their own ‘flavor’ in sound, so it would be best to audition them if you could. If in the US, a place like Magnolia/Best Buy may be a good place to visit. I’m sure there are many newer good models I am not aware of that may work for you. Personally, I would probably go towards the Parasound I mentioned, or as was also mentioned, a used Hegel would be hard to beat. Rega also makes a couple small form factor integrated amps that seems to get high marks from users, and are also in your budget-range.

Just have to make sure you have enough ‘grunt’ to drive your LS50’s easily and get the most from them. Current and damping factor are probably more important than pure watts.

And another new integrated I forgot to mention was the IOTAVX SA3. Pretty inexpensive, but highly rated.

Try and find a pre owned
Belles Aria like this


https://www.google.com/search?q=belles+aria&client=safari&hl=en-us&prmd=isvn&source=...
Enjoy 
JohnnyR

"...Nikko NR-719 tuner..."

Just to be clear that model Nikko is a receiver not just a tuner I think (Nikko made a couple of nice tuners back in the day, btw).
The LS-50s like a little juice so you want a powerful integrated amp. I'll suggest my normal pick, the Musical Fedelity M3si but better yet would be a M5si. You would be very happy with those. 

Thanks all! I’ll investigate. I should also say that I don’t need USB, digital, Bluetooth, etc. Just something simple for my turntable that will sound great. I'm interested in vintage options too...

lower end hegel unit

https://www.usaudiomart.com/details/649748343-hegel-h90/

stretching a little will be worth it in this case

Another would be the PS Audio Sprout.

Here is a link to a recent review, but he also provides alternatives which, in his opinion, sound better overall in that general price range.

https://youtu.be/qEASsQ06SCQ

That is a lot to ask in that price range, but the first one that comes to mind would be the Parasound Newclassic 200 Integrated Amplifier. New, it just a touch over $1,000 @ $1,200, But good luck finding one available. There is one used listed here for $900.