What amp for Polk LSiM 705?

Well it’s pretty inefficient so something with good control. 
 

never heard a Hegel but they are supposed to have great damping factor. Odyssey Stratos is a good one you can pick up cheap but not an integrated. I currently have a Misical Fidelity A1008 which was 6k 12 years ago for $1400. I really really like it. 
 

I would have to think most mid grade integrated amps would be a step up from your. Yamaha AV. Just look for something with good current and 100+watts at least. 

Topping PA5 for $349! State-of-the-art Class D 110wpc amp! Right behind the Benchmark AHB2 ($3500) for SQ! Why pay more!