By James Mackenzie
ROME, April 29 (Reuters) - New Italian Prime Minister Enrico Letta seeks the backing of parliament in a confidence vote on Monday, facing severe political and economic problems that will test the solidity of his broad coalition government in the months ahead.
Letta is due to speak in parliament at 3 p.m. (1300 GMT) before the lower house confidence vote in the evening, where he will be backed by his centre-left Democratic Party and former Prime Minister Silvio Berlusconi's centre-right People of Freedom.
A Senate confidence vote will follow, probably on Tuesday.
Sunday's swearing in of Letta's cabinet was overshadowed by the shooting of two policemen outside the prime minister's office in Rome. Nevertheless, financial market reaction to the end of months of political stalemate was positive with bond yields falling and shares rising.
Italy's cost of borrowing dropped to its lowest level since October 2010 at an auction of medium and long term bonds on Monday.
Letta was pushed into a coalition with Berlusconi after the centre-left fell short of a viable parliamentary majority in elections in February. He now faces a battle to maintain the unity of his government while passing unpopular reforms.
Berlusconi, who is fighting legal battles over a tax fraud conviction and charges of paying for sex with a minor, will not be in the cabinet himself but many people on the left find the idea of working with his centre-right party abhorrent.
In an interview on his own Canale 5 television station, Berlusconi said he hoped the left could undergo some 'self criticism' and learn from working with his party. He expected the government to last long enough to pass some vital reforms.
Berlusconi demands the scrapping of an unpopular housing tax and reimbursement of last year's contributions, a measure which would blow an eight billion euro hole in the 2013 budget. He also wants tax breaks for companies hiring young people.
It is not yet clear how Letta will handle this demand.
Berlusconi, whose last government was forced from office at the height of the euro zone debt crisis in late 2011, said he expected to play a leading part in shaping policy.
'As I am the president of the People of Freedom our representatives in the government will have continuous contact with our movement and with me,' he said.
On Sunday an unemployed man shot and wounded two police officers and a passer-by just as the cabinet was being sworn in at the presidential palace about a kilometre (half a mile) away.
Officials said the shooting, which the gunman said was originally intended as an attack on politicians, was an isolated incident but it was widely interpreted as a further alarming sign of public anger with lawmakers.
Italy's economy has been sluggish for over a decade with gross domestic product now lower than it was in 2001, companies stifled by high taxes and red tape and youth unemployment in some areas as high as 40 percent.
All this has fed into public bitterness directed at politicians, several of whom have been jostled or harassed by angry crowds recently.
Letta's cabinet, which includes a record seven women and Italy's first black minister, was shaped partly in response to disillusionment with political elites shown by the success of the anti-establishment 5-Star Movement in the last election.
But the party, whose fiery leader Beppe Grillo has built the 5-Star movement into the third biggest force in parliament by attacking traditional political elites, was singled out on Monday for criticism by his enemies.
The right wing daily Il Giornale, owned by the Berlusconi family - one of Grillo's favourite targets - carried the frontpage headline 'Il Grilletto' (The Trigger).
Letta has promised to address the social effects of the crisis and push the European Union away from its fixation with budget austerity and towards economic growth and investment.
With some doubt over whether his government will last a full five-year term, he is expected to try to pass at least a few basic reforms quickly including a change to Italy's much criticised electoral laws and a cut in the size of parliament.
Despite Monday's positive market response, there have also been notes of caution. 'The real tests will come in the next few weeks and months,' said Lorenzo Stanca, managing partner of Mandarin Capital, a private equity fund that invests in small Italian and Chinese firms. 'There is not much experience in Italy of a coalition government and it will be difficult.'
(Additional reporting by Danilo Masoni; editing by Barry Moody and David Stamp) Keywords: ITALY VOTE/
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