HOUSTON, June 29 (Reuters) - In not quite a month as Occidental Petroleum's ( OXY ) CEO, Richard Jackson has already been challenged with looking for ways to lift a lagging stock price and pay down more debt.

Longer term, Jackson may face a more fundamental question: whether to seek a buyer for the oil company that has a roughly $51 billion market cap.

Berkshire Hathaway ( BRK/A ), whose preferred stake costs Occidental hundreds of millions of dollars in annual dividends, already owns a quarter of the company.

Jackson, who first joined Occidental in 2003, took over the Houston-based company on June 1. He succeeded Vicki Hollub, who ran Occidental for a decade and engineered two major acquisitions that shifted Occidental's oil production heavily toward the U.S.

That positioning has proved advantageous as the U.S.-Israeli war with Iran rattled confidence in Middle East oil supply. Rivals like Exxon Mobil ( XOM ), with roughly 20% of its production in the region, were more exposed to disruptions.

Yet Occidental's acquisitions came at a steep cost and saddled the company with as much as $38.5 billion in long-term debt. Hollub reduced the debt to $15.2 billion by the end of her tenure, during which the share price fell 26%, lagging far behind its peers. Over the same time period, ConocoPhillips ( COP ) returned 153% and Chevron ( CVX ) returned 88%.

"The biggest opportunity is to clean up the capital structure, strengthen the balance sheet and increase shareholder returns," said David Byrns, a portfolio manager at American Century Investments, which holds an Occidental stake worth about $131 million, according to LSEG data.

During an earnings call in May, Jackson said his priority was to reduce principal debt to $10 billion in the near term, continue boosting free cash flow and grow oil production organically through technology.

"Richard has been spending time meeting with investors, hearing their points of view and reinforcing that our value improvement starts with executing from a strong balance sheet," an Occidental spokesperson said. GETTING OUT FROM UNDER BERKSHIRE

Occidental acquired Anadarko Petroleum for $55 billion including debt in 2019, aided by a $10 billion investment from Berkshire that requires Occidental to pay the conglomerate an 8% annual dividend. That is a higher payout than the typical junk bond now offers, and spurred criticism that Occidental was rewarding Berkshire much more than its other shareholders.

Occidental has paid off about $1.5 billion of the preferred stock and plans to begin redeeming the rest at a 5% premium when it is eligible to do so in August 2029.

Berkshire also owns 26.9% of Occidental's common stock, with warrants to buy $5 billion more until one year after Occidental redeems the preferred stock.

As a leader, Jackson proved successful at turning around a previously dysfunctional global drilling team, and is well-liked within the company, a former Occidental executive said.

Despite operational improvements so far, Occidental must either make more acquisitions or look for a buyer, said Bill Smead, chief investment officer at Smead Capital Management, which owns a roughly $201 million Occidental position.

The oil industry has seen a wave of mega-mergers in recent years, as producers sought to consolidate and lower operational costs.

"Either Occidental needs to get bigger and beef up the oil in the tank, or they're probably going to have to be part of a larger oil and gas company," Smead said.

Occidental and Berkshire should make clear whether they intend for Occidental to eventually become a subsidiary of the conglomerate, he added. Billionaire Warren Buffett, who was Berkshire's CEO at the time of its investment in Occidental, has said he did not plan to buy the company. Berkshire, whose new CEO is Greg Abel, declined to comment.

Berkshire's large stake limits interest in Occidental from potential acquirers, Smead said. "It keeps other investors from being aggressive."