How do you handle a vengeful ex spreading lies about you?

I see my ever-manipulative and truth-challenged ex is getting ever nastier, now falsely claiming that I "abandoned my kids". Yet she wanted to be with me because she knew how much I love my kids.

Hey Sydney- how many pics do you have of your baby daddy spending time with his kid? How much time do you even let him spend with his kid?

Unsurpringly, Sydney has also posted a lot of inaccurate info that completely distorts reality. The real reason our relationship didn't work out was her mental illnesses.

At any rate, rural Texan Sydney has no knowledge of Germany or any place other than her rural county so she has no idea of what it's like to live there.

-I am not a German citizen, yet Sydney bizarrely claimed I was. If she gets that wrong, what else would she get wrong?
-It's very hard even for people with good jobs to find any place to live in the area my kids reside. Rural Texan Sydney, who's lived with her father forever rent-free wouldn't understand that even if she were honest.
- I came to the US to make more money. I now do, since I am no longer living in the poor, rural county Sydney lives in, and she wanted me to work fast food or retail jobs. Lots of child support gets paid from that, right, Sydney?
-Sydney knows nothing about German family law, but she knows a lot about keeping a child from its father, which actually is what's happening to me right now.
-My kid's mom makes so much money that until recently I was not obligated to pay any child support under German law and she actually has to refund what she got for child support.
But I see people continually making excuses for the Pampered Princess, like "oh, but she's had it so rough". Oh sure, Papa only paid half of her expenses, real rough. Dozens of white knights she's always flirted with.

Sydney's behavior is consistent with someone who suffers from Borderline Personality Disorder. And they're known for making false allegations. How far will this vengeful person go?
How do you handle a vengeful ex spreading lies about you?
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