Elon Musk paraded his son X at a Formula One event in Texas in the middle of a custody battle with Grimes over where the boy primarily lives

• Elon Musk's current custody battle with Grimes hinges on where his children with the musician live.
• At a Formula 1 event in Texas, Musk was pictured waving to the crowd with his son X on his shoulders.

Tesla CEO Elon Musk made an appearance at Sunday's F1 race in Austin, Texas, with his son X perched on his shoulders.

The 3-year-old is at the center of a parental rights battle between Musk and his sometimes-partner, musician Grimes. Musk filed his parental rights lawsuit against Grimes in Texas last month, seeking to "establish the parent-child relationship" with the three kids he shares with the musician. A few weeks later, Grimes filed her own suit in California, seeking primary physical custody and joint legal custody of the three children.

At issue now in the case is where the children's primary residence is, since that will determine where the case will be heard. Grimes, whose legal name is Claire Boucher, said she moved to California with their one-year-old children Exa and Tau last year. However, Musk claimed the two children lived in Texas as recently as July. X, meanwhile, is "only absent from California over [Boucher's] objection," according to Grimes' October 6 request for Musk's case to be thrown out, previously obtained by Insider.

X has become a fixture at Musk's work events, attending live interviews with his dad, meeting with international leaders, and attending SpaceX rocket launches.

On Sunday, X was seen waving at the F1 crowd from inside Musk's Cybertruck, then later on his dad's shoulders in the pit lane. Photos also show the father and son watching the action from the Mercedes Garage at the event.

The boy's visibility lately brings up a key issue at stake in the current custody battle: jurisdiction, attorneys with expertise in family law previously told Insider. The case should go forward in whichever state the children actually live, so Musk and Grimes could potentially spend the next year just trying to prove to the court that the children live in the state they each filed in, the lawyers said.

Proving which state the kids live in could be an elaborate process that includes things like consulting flight logs and interviews with nannies. The judges in each state could also end up consulting with each other to determine the proper venue for the case to move forward.

There could be significant consequences related to which state the case actually plays out in. While it's unclear if the case will involve child support judgments, Texas caps monthly child support payments at $1,840 for one child, or $2,760 for three children, which is how many Musk and Grimes share.

That means despite Musk's billionaire status as the richest person in the world, the child support calculator in Texas won't spit out a value any higher than that. But in California, there's no cap.

So if Grimes successfully proves the kids live in California, it could turn out to be "one of the biggest child support cases of all time," Peter M. Walzer, a partner at Walzer Melcher and California family law attorney who often deals with out-of-the-ordinary financial disputes, previously told Insider.

Attorneys for Musk and Grimes didn't respond to Insider's request for comment.