* FTSEurofirst 300 steadies after Monday's drop
* Auto stocks rally as cheaper fuel boosts U.S. car sales
* Athens market closed on Tuesday but Greece still a concern
By Francesco Canepa and Sudip Kar-Gupta
LONDON, Jan 6 (Reuters) - A rally in auto stocks helped European shares steady on Tuesday, as uncertainty about Greece's future in the euro zone and worries about the implications of sliding oil prices dented broader market sentiment.
Strong December industry sales data from the United States boosted car makers with Fiat Chrysler Automobiles, the most exposed Europe-listed manufacturer, up 2 percent.
The STOXX Europe 600 Auto & parts index rose 1 percent, the best performing sector in Europe.
Auto sales are an early indicator each month of consumer spending, which is deemed to have benefited from falls in oil prices to a succession of 5-1/2-year lows.
While lower energy prices are widely seen as beneficial for European companies and consumers, they pile pressure on the finances of oil-producing countries such as Russia, raising the prospect of financial instability in emerging markets and hefty losses for investors in those regions.
"Whenever you see an asset class fall as sharply as oil did, it tends to unnerve people and that does have consequences, and you see that in Russia," Paul Sedgwick, head of investment at Frank Investments, said.
"But looking over history, a fall in oil prices boosts global growth. Oil will find a level, consumers will feel more confident and that will eventually feed through."
Brent crude briefly fell below $52 a barrel on Tuesday before staging a small recovery, which helped shares in oil majors Eni and BG Group rise over 1 percent after sharp falls in the previous session.
At 1217 GMT, the broad FTSEurofirst 300 index of pan-European shares was flat at 1,332.97 points.
Uncertainty ahead of elections later this month in Greece that have revived a debate about whether the country could leave the euro zone sapped appetite for risk.
The Athens stock market was shut on Tuesday for a public holiday, having fallen 5.6 percent on Monday.
"I don't think they will push Greece out of the euro zone, but just the sheer mention of this being a possibility is making investors nervous," said Darren Courtney-Cook, head of trading at Central Markets Investment Management.
Europe bourses in 2014: link.reuters.com/pap87v
Asset performance in 2014: link.reuters.com/gap87v
Today's European research round-up (Reporting By Francesco Canepa; editing by John Stonestreet)