WASHINGTON — In court documents that lawyers for Bain Capital sought to keep secret, the company and other leading private equity firms are depicted as unofficial partners in a bid-rigging conspiracy aimed at holding down the prices of businesses they were seeking to buy.

In Bain’s biggest acquisition, the $32.1 billion purchase of the hospital giant HCA in 2006, competitors agreed privately to “stand down” and not bid on the company as part of an understanding with Bain to divvy up companies targeted for leveraged buyouts, according to internal e-mails.

The documents have become part of a lawsuit in Federal District Court in Boston brought against Bain and other firms by shareholders who say the firms’ bid-rigging artificially deflated the sales price of more than two dozen companies and cost them billions of dollars.

Bain, founded by the Republican presidential nominee Mitt Romney, is a defendant in the lawsuit, which also names Goldman Sachs’s private equity arm and the Blackstone Group, the firm run by the investor Stephen A. Schwarzman.