NEW YORK, July 1 (Reuters) - New York senators must stay in Albany over the U.S. Independence Day holiday for extraordinary sessions, according to a decision on Wednesday by the governor, trying again to get warring lawmakers to pick a leader and finally enact bills.

Senate Republicans, saying they needed an "honest broker," responded by calling on Democratic Governor David Paterson to negotiate the power-sharing agreement he has urged them to work out with their Democratic colleagues.

It was just another twist in the leadership struggle launched on June 8 by a Republican coup. The resulting deadlock has prompted counties, cities and towns to warn that they will have to lay off workers and slice spending unless the Senate approves the routine bills they need to keep collecting nearly $2 billion of tax revenues and issue municipal debt.

Court fights, shouting matches, separate but rival GOP and Democratic Senate sessions -- and Paterson's vow to keep the state Senate in session until it enacts bills -- have all failed to bring forth a solution.

A highlight of Wednesday's session was Republican Senate Majority Leader Dean Skelos' plea to the Democrats, who at times have locked his party out of the upper chamber, to remove the podium guards.

"We are not going to rush up and take it over, anything like that," Skelos said. "Really, it is sort of unseemly and unbecoming that you have part of the militia up there."

Paterson told Albany reporters that his review of the past 40 years of deadlocked state and federal legislatures showed many of them resolved similar deadlocks by splitting leadership posts between the parties.

"There were none where two members of the rivaling party were in the same conference," he said.

The Republicans have rejected the Democrats' call for rotating the two leadership posts of Senate majority leader and Senate president between the two parties.

Spokesmen for the governor and the Democratic senators were not immediately available to comment on the GOP's call for Paterson to broker a deal. Paterson previously has rejected serving as a power broker as at least one of his predecessors has done. (Reporting by Joan Gralla in New York; Editing by Jan Paschal)