AMR has one too...here's a quote from a random press release I found on Google.

"American Maintenance Services offers a full line of airframe, engine and component, and line maintenance services, customizing those services to meet the specific needs of the client. American's MRO business has 62 customers -- 80 percent of which are repeat business -- located in North and South America. Services are provided by American's overhaul bases in Tulsa, Okla.; Fort Worth, Texas; Kansas City, Mo., and at line maintenance locations in the United States, Latin America and Europe."

We sometimes see local media about these guys, since they use the old TWA overhaul base at MCI. The business pages usually give 'em a write-up whenever they close a big contract with another airline to do overhaul work.

It wouldn't surprise me if United and American have the largest maintenance divisions, but honestly I'm not really sure. There are probably some third-parties out there (unaffiliated with a legacy airline) that do a lot of this work too.