Prince Jacques and Princess Gabriella Hold Signs for Princess Charlene as She Seeks Treatment

In the wake of the exclusive news reported by PEOPLE that Princess Charlene is receiving treatment in a facility "outside of Monaco," the royal family gathered on Friday to celebrate Monaco's National Day. But the festivities were bittersweet this year as Charlene's 6-year-old twins appeared on the balcony holding signs of support for their mother.

The traditional ceremony, which sees the entire assembled Monaco royal family traditionally greet their citizens, is typically capped by a balcony wave by the royal family from the palace. As Prince Albert and his children waved to the crowd below, Prince Jacques and Princess Gabriella, who will turn 7 on Dec. 10, disappeared from view and reemerged with handmade drawings they made for Charlene. Gabriella's sign read: "We Miss You Mommy," and Jacques' read: "We Love You Mommy."

"Prince Albert said he wanted today to be as close to a normal National Day as possible," a palace source tells PEOPLE. "And in particular, as much as can be for the children."

Prince Albert exclusively tells PEOPLE that the decision to seek medical care just days after her return to Monaco after six months in South Africa was made jointly by the couple, assisted by Charlene's brothers. Albert says the 43-year-old princess is suffering profound "exhaustion, both emotional and physical." Dismissing a variety of rumors, he says her recovery will require clinical care lasting several weeks at minimum.

With their mother away, Jacques and Gabriella's special bond was on display. During the courtyard presentation of medals (which was made longer this year due to last year's cancelation amid the pandemic), the duo watched closely alongside their aunts Princess Caroline and Princess Stephanie. At one point, Jacques reached over and sweetly held his sister's hand. Gabriella then reached for her brother's hand moments later.

Gabriella, who is already taking after her late grandmother, Princess Grace, was adorable in a white lace dress from Tartine et Chocolat under a brown Brunello Cucinelli coat and pink beret. Mom Charlene also wore a beret to the 2019 National Day celebrations. Prince Jacques, meanwhile, suited up in his Carabinieri ceremonial uniform, which he debuted in 2019.

This year's outdoor ceremony was intended to be a welcome home celebration for Charlene and included several elements commemorating the couple's 10th anniversary. The princess had been "grounded," as she previously put it, in South Africa since early May due to an ENT infection which necessitated a number of corrective surgeries over her six-month absence.

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Charlene's prolonged absence from Monaco caused the princess to miss her 10th wedding anniversary and gave rise to numerous reports concerning a rift in the couple's marriage as well as speculation concerning the true nature and degree of her medical issues. The couple consistently denied these tabloid reports.

"Because I know there are rumors out there, let me say: This is not COVID," Albert tells PEOPLE. "And it's not cancer-related. It's not a personal relationship issue. And if you want to discuss another speculation, it isn't related to plastic surgery or facial work at all."

Her current state, he says, is the result "of several factors which are private matters." He adds, "she was clearly exhausted, physically and emotionally. She was overwhelmed and couldn't face official duties, life in general or even family life."

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Jacques and Gabriella "knew that she was tired. That she was not quite herself," Albert continues. "And then I told them that she needed some time to rest, and that she is now somewhere not too far away, that we can go see her one day. They understood that.

"They know that she's going to talk to them very soon. They miss her, of course, but they understand. Kids at that age they understand."

As for how he's faring, he says, "This is usually the busiest time of year. I don't have time to be going around with a too-sad face. I'm okay. Of course, I'm saddened by the situation, but I know she's in a good place now. That she'll recover and that this is the best place for her."