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The House returns Wednesday from a nearly two-week Independence Day vacation. Meanwhile, the Senate canceled its own July 4th vacation, scheduled for this week, after Mr. Obama and some Republicans demanded the chamber stay in session to work on a deal.

Senate Majority Leader Harry Reid agreed to those demands late last week, but rather than the budget, he scheduled votes on a resolution authorizing the president to continue with U.S. military action in Libya.

On Tuesday afternoon, under fire again by Republicans, Mr. Reid pivoted once more, announcing the Senate will put off the Libya votes and instead debate a resolution Democrats have written that calls for wealthy taxpayers to be a part of “shared sacrifice.”

The measure is nonbinding, so even its passage would amount only to a symbolic step.

The resolution calls for those earning $1 million or more to “make a more meaningful contribution to the deficit-reduction effort.”

Mr. Reid, Nevada Democrat, said he expects Republicans to offer their own alternatives.

Meanwhile, Republicans in both chambers are preparing for votes later this month on a constitutional amendment requiring the government to balance its budget. That will offer yet another vehicle for a debate over long-term debt.