A New Integrated Amp

I'd advise getting the speakers first. 

Then on integrated amps. They get much better in the $10K range. So, I would consider used from a company like Pass. Best to match the electronics to the speakers. Best to have the speakers set up properly and broken in before you start looking for the amp. You want the amp to complement your setup and you will have a better idea of what sound you want after you have the speakers. 

+1 @ghdprentice 

Amp first usually are those who are very familiar with the sound of the amp, often a boutique amp such as Kondo, Shindo, etc. Otherwise it’s usually best and easiest to start with speakers first.

Speakers first one goes out to demo speakers typically at audio stores and audio shows to find the sound that personally resonates with them, then take notes of the accompanying electronics to help decide what direction to go

+2 ghdprentice and kennyc

Go up in quality  Alluxity audio ,parts of Denmarks Vitus brand for under $11k

a o excellent integrated that meets or beats pass labs and Luxman ,Accuphase.

it’s a giant step up them the brands mentioned .

I would suggest finding your speakers first.  The more efficient they are, the more integrated amp choices you will have.  I got nice sound from a rega, ps audio and rogue pharaoh with dali’s and jm reynauds.  The rega hit its ceiling with my magnepans.  The ps audio and rogue pharaoh handled the magnepans better.  I tried an nad amp on the magnepans but the sound lost musicality.  I did also have good results with a vinnie rossie integrated on all the speakers.  Component matching and room treatment go a long way.