U.S. oil refiner Phillips 66 has retained two top financial and legal advisers for its duel with activist investor Elliott Investment Management, according to a person familiar with the matter.

Elliott wants two board seats to be filled by executives with refining experience and faster action on restructuring the fourth-largest U.S. oil refiner to improve its lagging financial performance, the New York activist said on Wednesday. It disclosed a $1 billion stake and took public its calls for change.

Phillips 66 is relying on financial and legal advice from Goldman Sachs and Wachtell, Lipton, Rosen & Katz, the person familiar with the matter said.

Shares in the Houston refiner rose for a second day as investors cheered the activist's proposal for refining cost cuts and a refreshed board of directors.

The stock traded at $127.95 at mid-day Thursday, up 8.4% from its $118 per share close prior to Elliott's going public with its letter.

A Phillips 66 spokesman declined to comment on the advisers or whether it has new meetings scheduled with the activist. The company has acknowledged it held and will continue those talks but has not said whether it would accept Elliott's director candidates.

"We think some portfolio optimization will be necessary to achieve full valuation, as the conglomerate structure hides the true value of PSX's assets," wrote Tudor Pickering Holt & Co. refining and chemicals analyst Matthew Blair. He rates the shares a "buy" with a $130 target price.

The company could divest some non-operated pipeline properties, a European retail fuels unit and its 50% stake in CPChem, a chemicals joint venture with Chevron, Blair wrote.

(Reporting by David French in New York, additional reporting and writing by Gary McWilliams; Editing by Chizu Nomiyama)