Oksana Grigorieva is not publicly commenting about statements Mel Gibson allegedly made about her to screenwriter Joe Eszterhas, but her rep is doing the talking, and he tells TMZ authorities should launch a probe.

Eszterhas claims Mel talked about offing Oksana several times. To that Oksana's rep, Steve Jaffe says, "Mr. Eszterhas' charges are extremely serious and should be investigated by the authorities immediately."

Jaffe says Oksana has no comment. We're told she signed a confidentiality agreement with Mel when they ended their custody/money war.

Mel Gibson has done all he needs to do to prove he has conquered his anger issues ... TMZ has learned.

Mel's lawyer, Blair Berk, appeared before Judge Stephanie Sautner this AM and showed her honor proof that Mel not only completed the required number of hours of court-ordered anger management therapy -- he exceeded it.

As a result, all Mel has to do now to get off of probation -- for pleading no contest to battery against Oksana Grigorieva -- is keep his nose clean for 2 more years, and then he's officially off informal probation.

Oksana Grigorieva performed live in Hollywood last night and while the place was packed, Oksana's performance was ... well you can listen for yourself.

Oksana played the Gardenia Lounge in West Hollywood last night as she tried to further the singing career that was once helmed by ex-boyfriend Mel Gibson. He was her executive producer at one point and when they split, securing the rights to her music was Oksana's mission.

As for the show, our guy inside said the place was quite crowded and everyone was there specifically to see Oksana.

Afterwards, she told our guy the show went well.

Oksana Grigorieva -- the woman who became famous when Mel Gibson hooked up with her and then tried executive producing her very first album -- IS GOING TO HOLLYWOOD!!! ... TO SING!!!

Oksana has been booked at a legendary Hollywood lounge to perform next Friday and Saturday night. The lounge -- Gardenia -- has been around for years, and featured some big performers.

It's interesting -- Oksana was hell-bent on getting the rights to her music from Mel during their breakup. Oksana secured the rights and now is trying to make it on her own.

Check out the sample from the Mel-funded album (above) ... and then you be the judge.

If Mel Gibson got a grade for his probation, he'd score an A+.

Mel was in Judge Stephanie Sautner's court today for a progress report in his battery case -- remember Oksana Grigorieva?

Judge Sautner said she was more than impressed that Mel has completed twice the counseling requiring of him, and he was right on track on his 12-step program.

Mel pled no contest in March to misdemeanor battery.  He's currently on informal probation.

Oksana Grigorieva got in the Thanksgiving spirit yesterday in L.A. -- handing out food at the same homeless shelter where Kim Kardashian and Jennifer Love Hewitt cooked up a real mess.

Everything's just gravy for Mel's ex.

Mel Gibson is back in the game ... sucking face with a woman who looks an awful lot like that Russian singer with a mean iPhone.

Mel and his chick hit up L'Ermitage Hotel for a romantic dinner earlier this week.  When they left Mel went in for the kill.

A few days before Mel went back to the scene of the crime -- Moonshadows -- where he went just before he was arrested in 2006 and went on that anti-Semitic rant.  Mel was there with another mystery woman and it was all very uneventful.

Everything seems to be comin' up roses for Mel.


TMZ has learned the mystery woman is "Alphas" star Azita Ghanizada. Her rep tells us it was NOT a date and the dinner was not romantic in nature.

Like a 13-year-old Mike Singletary ... Paris Jackson was a focused instrument of destruction as a linebacker during a flag football game in L.A. yesterday.

Jackson -- who's the very first female in the history of her private school to join the squad -- also played a little offensive line ... and got STRAIGHT UP PHYSICAL with the boys, throwing some pretty aggressive blocks.

But Paris wasn't the only celebu-spawn on the gridiron ... Oksana Grigorieva and Timothy Dalton were standing on the sidelines to support their son, Alexander, who's also on PJ's team.

Jackson's school emerged victorious ... and after the game, she tweeted about it ... saying, "We Won 32-6!!!!!!!! Yeahhh Boiiiiii!"

Kristian Otto Herzog -- the guy who was once at the center of the bitter battle between Oksana Grigorieva and Mel Gibson -- is now at the center of the race for the Republican presidential nomination ... alongside Herman Cain.

Herzog was spotted on Friday with the GOP candidate during a campaign stop in Detroit, MI.

Cain's spokesperson, J.D. Gordon, tells TMZ this was Herzog's first job with the campaign and he worked with a team of 2-3 guys. Our sources say they don't believe Herzog is scheduled for any future events ... nor do they think he will be asked to provide future services.

Herzog -- a convicted felon with a penchant for impersonating police officers -- worked as Oksana's bodyguard and claimed to have had a sexual relationship with her. He sided with Mel when she accused him of domestic violence.

Oksana Grigorieva just made a beeline for family court after her settlement with Mel Gibson hit a snag ... TMZ has learned.

As TMZ first reported, Oksana and Mel struck a deal in which Oksana would walk away with $750,000, the house she's living in, and some other stuff. In return, she'll leave Mel alone.  They also agreed to 50/50 custody of Lucia.

The settlement agreement is scheduled to go before the family court judge tomorrow for final approval, but we're told the lawyers are still fighting over how it's worded and they've reached a stalemate.

Our sources say Mel's lawyers asked the judge for more time so they can hash it out, but Oksana showed up to court this morning to fight the request ... because she wants the deal done STAT.

Judge Scott Gordon gave Mel and Oksana an extension to sort out their differences -- but only until November 1st ... less time than Mel's people were hoping for.

Mel Gibson is taking issue with legal documents filed by Oksana Grigorieva's lawyer -- which state that Mel paid $100,000 to Oksana's son Alexander Dalton, the son she had with actor Timothy Dalton.

Mel's rep tells us, "The deal between Mel and OG is exactly what was set forth on the record, in open court, on Augst 31, 2011" -- when Mel agreed to pay Oksana $250,000 as part of their custody settlement agreement.

His rep continues, "Mel is paying exactly the sums set forth to OG -- and is paying no additional sums to Alexander Dalton or anyone else."

"How OG chooses to have any sums due from Mel allocated is her business."

But as we previously reported, TMZ has obtained legal documents filed by Oksana's lawyer -- which claim, "Of the first $250,000 paid ... $100,000 belongs to Alexander Dalton who is waiving his rights to sue Mel Gibson and must be compensated for this to be lawful."

The docs continue, "If Alexander Dalton does not get his $100,000.00, he cannot waive his rights to sue. Mr. Gibson negotiated for that as a part of the settlement and is entitled to the benefit of his bargain."