* Risk appetite recovers, Bunds dip, but tension remains high
* German Bund auction yield at lowest since July 2012
* Spain, Italy rebound as some keep faith in ECB backstop
By Emelia Sithole-Matarise and William James
LONDON, March 20 (Reuters) - German Bunds dipped on Wednesday as some investors trimmed exposure to low-risk debt on bets that policymakers will find an eventual solution to Cyprus' debt problems.
Cyprus sought a new loan from Russia to avert a financial meltdown after its parliament rejected on Tuesday plans to partially fund a European Union bailout for the indebted island with a deposit levy.
While German Bunds gave back some of this week's gains, few in the market expected a significant fall in the euro zone's least risky bonds given uncertainty about Cyprus' future in the currency bloc and fears of a run on its banks when they reopen.
"Bunds have had a good run so some people are thinking they might as well cash in. Generally the market probably feels that if Russia comes in and gives them some money that eases the initial problems," said Alan McQuaid, chief economist at Merrion Stockbrokers.
"I don't think there's going to be a meltdown but there are still more questions than answers ... Until this is resolved and there's greater clarity on Cyprus, demand for core bonds will remain strong near term."
Underscoring the caution, a new German 10-year bond found strong demand compared to similar sales this year and recorded the lowest auction yield since July 2012 - a time when an investor exodus from the region's lower-rated states threatened the euro zone's future.
In the secondary market, 10-year German bond yields were 4 basis points higher on the day at 1.39 percent, retreating slightly from Tuesday's 2013 low of 1.34 percent. German Bund futures shed 32 ticks to settle at 144.30, but this was about a quarter of their 120 tick gain over the last two days.
The precarious position of Cyprus's banks, which are heavily dependent on European Central Bank support, has increased the urgency of finding a new plan. Without a solution, Cyprus risks default as early as June, when a 1.4 billion euro bond matures.
Cypriot banks remain shut and traders said many investors were waiting to see whether fears of a rush to withdraw money from the system would materialise. Any signs of a bank run would accelerate the sense of crisis and switch the focus to Italy and Spain for any signs of spillover.
The chaos in Cyprus has already caused investors to back away from bonds issued by the region's other struggling countries, and upward pressure on Spanish and Italian bond yields was expected to remain high.
"The risk is that we see more serious contagion... we're sticking to our long Bunds, short periphery positions," a trader said, referring to bets that the yield gap between German and weaker bonds would widen.
Despite the expectation of more pressure, a rebound in equity markets and expectations that the ECB's safety net would limit contagion helped keep Italian and Spanish yields in check.
Ten-year Italian yields were 7 basis points down on the day at 4.64 percent while the Spanish equivalent yield was 4 bps lower at 5.0 percent.
"On the long term I would say all the widening we have seen in Italy and Spain are opportunities to play for another spread compression," said BNP Paribas strategist Patrick Jacq.
"But, having said that, it's clear that the very near-term environment remains risky enough not to enter these kind of trades massively."