Looking for suggestions on speakers and an integrated amp for a second system.

My main concern is sounding full at low volume level.

Out of favour in amplifiers for quite some years, but when understood correctly a "loudness"  feature is useful for fullness at low volume. Vintage rocks!

It shouldn't be used at volumes other than fairly low or weirdness happens.  There was some discussion here within recent memory.

Here's a link to a technical discussion. including recent developments.

@noske Yamaha still has this feature in some of their Mid-Fi Integrated's

For fuller sound at lower volumes, I'd suggest using part of the budget for a small subwoofer.  I have run one in my secondary home office rig for years and it allows me to have great performance at any volume but also the ability to run bookshelf speakers on stands, making the room feel larger not as cluttered, not having to worry about pulling larger towers out into the room to breathe, etc. 

I am thinking the loudness control would be helpful. May eliminate (maybe not entriely) the need for a subwoofer.

Yamaha and Vincent made integrateds with that feature. I am sure there ore others. 

Look for speakers with wide dispersion. Ones that serve an area larger than the lisening chair. 

This may not sound very "audiophile", but give some thought to a tube integrated like Line Magnetic, Rogue, ARC, etc. and some Heritage Klipsch speakers - some older Heresy or Quartets or Fortes.  You can often find the 1 and 2 versions of these well under $1K then put the rest into the integrated.

Most people think of Klipsch speakers as "rock speakers" and that they are bright and unrefined.  That's true (IMO) when they are played loud with rock, but they can be quite special when played at reasonable volumes with female vocals and acoustic instruments, especially guitars.

I second the recommendation for a small subwoofer.  Something like a Rythmik L12 or a SVS SB-2000 will add presence to the music.