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

If you have the space for Magnepan speakers I would go with the 1.7 and a Parasound Hint Halo integrated I had this system in my a  12 x 15 room and it sounded terrific. Used this combo should be fit your budget.

there are many many terrific choices to choose from with a US$3700 budget

what are your sources? vinyl? streaming? cd’s? ...

is this a dedicated listening room where the speaker placement can be without compromise for aethestics?  if not what are the restrictions or cosmetic limitations?

Just curious: What source component(s)? Digital? Analog? Turntable? Do you want/need a phono stage or DAC in the integrated?

I will be using digital through a wired4soud dac2 did se.

I put together a pretty good digital system for my office about the same size as your room, that meets your budget.  The system sounds great, almost as good as my main rig but really, with the bookshelf speakers, its only for relatively near field listening.

It consists of a Pro-Ject S2 Ultra Streamer, a Pro-Ject S2 Digital Pre/Dac, and a Pro-Ject Amp Box.  I used all my 'second' cables (i.e. those I replaced for better cables) which are OCC 6n silver-plated speaker wire (ELF Silver Surfers), a Wire world USB (ultra-violet) and bespoke LPS units for the Streamer and Amp.  Three of the pieces were pre-owned.  I use the steamer as a Roon endpoint from my Innuos Zenith Mk3 Roon core, but the Pro-Ject software works fine and lets one use it to stream Qobuz or Tidal (possibly others) without Roon.

My main concern is sounding full at low volume level.

IME, that's a hard one, pal.

I have that problem in my office system and use the fairly high efficiency single driver, fostex based, Brines 2000.  They are no longer made, but single driver is a thought, and there are numerous options out there I bet would sound pretty good at low db.

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.

If I were starting from scratch with that budget I would look at Kef LS50 wireless speakers and mate them with a Rythmik Audio F12 sub (or maybe look into REL or even Kef's own subs but I think the Rythmik will give you more bang for the buck).

I am quite familiar with the LS50 sound (paired with a REL T/5 sub).  You are probably not going to find anything in the $2500 price range (estimate for the Kefs) that comes close (speakers plus amp and all the cables, etc.).  That will give you $1200 to find a decent sub(s).

You can put a lot of fantastic systems together with $3,700. I'll throw out one option; get a Belles Aria integrated and a pair of Vandersteen 1ci speakers. You could be happy with this system for years to come. Good luck!