Need integrated for Tyler linbrook Signature

For a solid state integrated, I can hightly recommend the NAD M3. It is powerful and in my setup was somewhat analog, definitely not bright. If you feel the need to use tone controls on some recordings it provides those. I never used them on my M3 but some may like that feature. It also has many other customizing features that most other high end integrateds lack. If the look of componentry interests you, it is a beautiful integrated. It's loaded with flexibility and features for various applications. Unlike the other pieces NAD has made (besides possibly the Silverline), in my opinion, this one is high end. There is one available here now.

Might want to try a Portal Panache - use one with Linbrook System 2

I demoed my Jadis OR on the LS's and had no problem getting bass good response. These speakers are definetly tube friendly, provided the amp has high quality trans as does Jadis.
That would Cayin. Ck out their many models. Maybe the new Cayin 100, with 8 KT88 tubes. Abit over 2K but worth looking into. Besides you have a top class speaker, well worth over the asking price (fair value is $15K, Ty sells them for 1/3 the price), so you need a first class amp.
There's not many first class amps in the 2K range.
good luck

How about a BAT integrated.

I am driving mine right now with a VK200 and they sound great. I also have a Levinson No.29 at 50wpc and it works very well with my Linbrook Sigs.