* Euro rises on speculation euro zone will remain intact * Bonds fall on profit-taking, hedge against debt solution * Stocks falter, then fall on still-gloomy economic data By Herbert Lash NEW YORK, June 4 (Reuters) - The euro rallied and bonds retreated from last week's record low yields o n M onday as speculation increased that authorities will seek greater fiscal integration in the euro zone. U.S. stocks fell, but oil, copper and other commodities rebounded as investors speculated that new action may be in the works to address the debt crisis and keep Greece from leaving the euro zone. A rally in Europe's troubled banking sector lifted battered Spanish, French and Italian stocks, with the euro zone's blue-chip Euro STOXX 50 index closing up 0.5 percent. An index of the euro zone banking sector rose 3.4 percent. The euro was 0.4 percent higher at $1.2480, off the near two-year lows hit o n F riday. German Chancellor Angela Merkel is pressing for much more ambitious measures, including a central authority to manage euro-area finances and major new powers for various European entities. In Spain, Prime Minister Mariano Rajoy is pushing for a direct European rescue of the country's troubled banks. France and the European Commission signaled their support on Monday for an ambitious plan to use the euro zone's bailout fund as European officials try to reassure investors they can contain an escalating crisis. Senior European Union officials have promised decisions at a summit at the end of June to resolve the 2-1/2-year debt saga to deepen integration in the euro zone and underpin the common currency, showing they are committed to its future. Stocks on Wall Street zigzagged between losses and gains. The Dow Jones industrial average was down 34.70 points, or 0.29 percent, at 12,083.87. The Standard & Poor's 500 Index was down 3.00 points, or 0.23 percent, at 1,275.04. The Nasdaq Composite Index was up 1.03 points, or 0.04 percent, at 2,748.51. In thin European markets, the FTSE Eurofirst 300 index of top shares closed down 0.5 percent at 949.94 points. The MSCI world equity index fell 0.4 percent to 290.81. Traders took profits in safe-haven U.S. and German debt, wary that a policy response to the euro zone's debt crisis might be in the works. "It's relatively difficult to be positive on these developments," said Marius Daheim, senior fixed-income analyst at Bayerische Landesbank. "But we haven't given up because the past has also taught us that European politicians usually move when things become really dangerous. I think we are quickly moving toward this point." The benchmark 10-year U.S. Treasury note was down 19/32, the yield at 1.5223 percent. The price of the 10-year German bond fell and its yield rose to 1.215 percent. Another factor on traders' radar was that potential monetary easing may come from a meeting of the European Central Bank on Wednesday, as some investors positioned for an outside chance of a rate cut. Factory prices held steady in the euro zone in April, giving the ECB some room to cut rates. "They (the ECB) have made it clear that they want the solution to come from Europe's leaders, but the recent deterioration in economic data and slide in asset prices makes easier monetary policy inevitable," said Kathy Lien, director of currency research at GFT in Jersey City, New Jersey. New orders for U.S. factory goods fell in April for the third time in four months, the latest worrisome sign for the economic recovery. Oil prices snapped a string of four lower finishes as a drop to multi-month lows attracted bargain-hunters and as the euro rose against the dollar on hopes that Europe's leaders can keep the euro zone intact. Brent crude futures rose 42 cents to settle at $98.85 a barrel. U.S. July crude rose 75 cents to settle at $83.98, after earlier falling to $81.21, the lowest since prices were last under $80 a barrel in October. Spot gold prices fell $7.20 to $1,618.40 an ounce.