The merry go round. $700 seperates or integrated?

I am familiar with the RS 2.5 and other Infinity speakers from that time, and I think it needs a very powerful amp. Infinity recommended a minimum of 100 watts, and that should be a very high current amp. Being a 4 ohm speaker dropping much lower and the type of drivers used means this speaker will be a very demanding load on the amp. It would be best for the amp to double power into 4 ohms or at least get somewhat close to a double. I seriously doubt that many of the amps listed in this discussion could adequately drive that speaker. Not meaning that anything is wrong with those amps, they just don’t have enough power. Like putting a Toyota Tacoma 4 cylinder engine in a Hummer H2 resulting in much less overall  performance of the Hummer and it being capable of more.

The issue is not integrated or separates, but making sure the amp has adequate power to drive and control the speaker.

An Adcom 555 puts out 200wpc at 8 ohms. I would hope that would be enough.

I second the Audio Refinement integrated with a dedicated headphone amp. Both can be had for a total of $700. 

"I second the Audio Refinement integrated"

I'm sure the Audio Refinement is a nice integrated amp, however it would be a very bad choice for the OP speakers, Infinity RS 2.5. Excerpt from Stereophile review,

"the amplifier was current-limited into lower loads"

As I previously stated, the speaker is 4 ohms dropping much lower, and a very demanding load. It needs high current that is not limited.

Pretty sure a Cambridge 840A V2 will drive your speakers no problem. If you have a Cambridge dealer near you, see if they will let you drag home the newer 850A; if you like it then find a 840A.

Know Adcom amps well, especially the older units and their preamps. Unless you're going to spend some money to mod one, such as with Musical Concepts or if you know how to mod yourself a CA 840A V2 is a much more refined amp. 

If you're going to drift off to separates as some are suggestion I really don't think you're going to outclass a newer / used integrated. There's a lot of really nice older amps out there that are under your budget but it's now finding a decent preamp for what you have left for funds. If you don't get the best preamp you can for the money, you're wasting your time and money on an amp. 

If you can double your budget I agree you can likely build a better system with separates. 

The CA 840A V2 is not overly polite, laid back unit; so if you're wanting that look on. It is lively and has PRAT. Not bright or edgy but is revealing . Use decent cables, and PC and it will reward you. Very well built. When I had mine, I had B&K, Counterpoint separate amps as well and the 840A was much better. My speakers at the time were Apogee Centaurs which are fairly revealing speakers with their ribbon tweeters. 

Eventually I did replace my 840A but it took a ModWright 9.0 SWL and an Audio Research 100.2 to do it. I dragged home a lot of gear to outdo it; I'm lucky I have two great stores nearby that let me take stuff home. Best advice any of us can give you is try before you buy.