The Supreme Court is set this week to begin announcing decisions on roughly a dozen remaining cases, including contentious ones on abortion and immigration.

The rulings will be released from the bench starting Monday, with the high court typically waiting until the last week of June to announce those on the most contentious issues, in a tradition known as "spring flood" and just before the justices go on summer recess.

Rulings on this season’s slate of cases were complicated by the death of conservative Justice Antonin Scalia -- leaving the bench with four remaining conservative justices and four liberal ones.

"Eight, as you know, is not a good number for a multi-member court," Justice Ruth Bader Ginsburg recent said. "That means no opinions and no precedential value; an equal division is essentially the same as a denial of review."

Such ties mean a default victory for the party that won at the lower court, but only in states covered by that appeals court's jurisdiction.

The lack of precedent leaves legal uncertainty nationwide. And the high court could re-argue the case when it operates at full strength with nine members.

The court has already deadlocked twice this term, and essentially punted in another high profile case.

A challenge by religious-affiliated employers -- including a Catholic charity of nuns -- to ObamaCare's contraception mandate was sent back to the lower courts.

"The Court expresses no view on the merits of the cases," the unsigned ruling read.

The outcome suggested the court lacked a majority for a significant ruling and that it could be a template for other divided cases to be resolved in the coming days.

On a pending ruling regarding abortion-clinic restrictions in Texas, Carrie Severino, chief counsel of the conservative Judicial Crisis Network, see a possible 4-4 decision but thinks the case will come before the court again, in perhaps a year or two.

However, some progressives hope the court's so-called "swing vote" will break with his conservative colleagues.

"I think all eyes are going to be on Justice [Anthony] Kennedy and I don't think it will split 4-4," said Elizabeth Wydra, president of the Constitutional Accountability Center. "I think Justice Kennedy will join the more liberal justices to say the Supreme Court has reaffirmed there is a constitutional right for a woman to choose whether or not to have an abortion."

Both court watchers spoke Sunday to Fox News Chief Legal Correspondent Shannon Bream, on "America's Election Headquarters."

Scalia’s unexpected death in February, during a presidential election year, left the court shaken and short-handed, a situation Ginsburg noted wistfully.

"The Court is a paler place without him," she said of her best friend on the court.

President Obama has nominated Merrick Garland as a replacement, but Senate Republican leaders have vowed that the federal appeals court judge will not get a hearing or floor vote this year.

That would leave his successor with an immediate opportunity to shift the court's shaky ideological balance to either the right or left.

While public debate over the direction of the nation's highest court continues to animate the presidential race, the court remains notoriously tight-lipped on internal discussions.

Still, Chief Justice John Roberts last month told a group of judges he and his colleagues were operating under a new dynamic.

"I try to achieve as much consensus as I can," Roberts said. "We kind of have to have a commitment as a group. I think we spend a fair amount of time-- maybe a little more than others in the past-- talking about things, talking them out. It sometimes brings you a bit closer together."

Another consequence of the current vacancy is that the court has accepted fewer cases for full review, meaning its fall docket is so far pretty thin.

Under court rules, it takes at least four justices to put a case on the argument calendar, and five to ultimately prevail on the merits.

But Roberts said a go-slow approach often is best long term.

"I think we should be as restrained in when we decide the issues when it's necessary to do so," he said. "I think that's part of how I look at the job."

Among the cases yet to be decided:

AFFIRMATIVE ACTION: Fisher v. University of Texas at Austin. The case could be a definitive, landmark ruling on the use of race in public university admissions. Justice Elena Kagan has already recused, meaning conservatives might have a 4-3 edge to cut back on such policies. Or Kennedy could side with his more liberal colleagues to uphold the school's diversity model.

The justices first heard the case three years ago but essentially sidestepped a binding resolution. The court at the time affirmed the use of race in the admissions process but made it harder for institutions to use such policies to achieve diversity. The 7-1 decision avoided the larger constitutional issues.

ABORTION ACCESS: Whole Woman's Health v. Cole. The constitutionality of a Texas law requires all clinics performing abortions in the state to be "ambulatory surgical centers" and that doctors performing abortions first obtain admitting privileges at a nearby hospital.

Lawmakers in the state's Republican-majority Legislature have said the regulations contained in the 2013 law -- known as H.B. 1 -- would improve patient care and safety.

Abortion rights groups argue the law is designed to make it nearly impossible to operate an abortion clinic in Texas. Only 10 such health centers would qualify to stay open, and large areas west and south of San Antonio would have no full-time abortion providers.

EXECUTIVE POWER: U.S. v. Texas. The case is a multi-state challenge to the president's use of executive power to block certain undocumented aliens from being deported, despite being in the country illegally. A tie decision would essentially be a win for the states, since the lower courts have put the president's programs on hold.

At issue is whether as many as 4 million immigrants can be spared deportation -- including many who entered the U.S. illegally as children, or those who are parents of citizens or legal residents.

The president's programs -- known as the Deferred Action for Parents of American Citizens and Permanent Residents (DAPA) and Deferred Action for Childhood Arrivals (DACA) -- effectively went around the Republican-led Congress. Opponents -- including 26 states and GOP members of Congress -- say the White House's broad immigration policy plan exceeds constitutional power.

PUBLIC CORRUPTION: McDonnell v. U.S. The case addresses the definition of "official acts" that can lead to being prosecuted for public corruption. Virginia's former GOP Gov. Bob McDonnell was convicted of providing favors to a friend and benefactor.

In oral arguments from April, a majority of justices appeared poised to void McDonnell's conviction and place new limits on the reach of federal bribery laws.

Several on the court, across the ideological spectrum, expressed major concerns that the laws give prosecutors too much power to criminalize the everyday acts that politician perform to help constituents.

SEARCH AND SEIZURE: Birchfield v. North Dakota; Bernard v. Minnesota; Beylund v. Levi. The justices are considering the constitutionality of warrantless roadside drunk-driving tests and of assumed consent to such tests.

The high court also expressed doubts in oral arguments about laws in at least a dozen states. Drivers prosecuted under those provisions claim they violate the Constitution's 4th Amendment protections. Some states suggested it would be too burdensome in many instances to seek an expedited search warrant every time.