What us the red star next to the gene trying to indicate?

https://www.sylestia.com/view/pets/?petid=7840569

Hi, the red star next to the gene means it has a restricted wild trait from the Lost Grove. In the case of your particular pet, it has that wild trait as a hidden one.
Here's a post that was made about these traits sometime back:
https://www.sylestia.com/forums/?thread=91213

If the trait were to be visible on your pet, it would look like this;

Can you achieve ir through breeding?

If you have a pet that has the trait either hidden or carried, I'd assume you could achieve it through breeding if the trait were to be bred to a pet that either has the same matching carried or hidden trait, or a pet that has it visible.
Another way may be to breed to a pet that has a trait that's lower down than the restricted wild trait(Like for example, in the case of your pet, there are Lucky Runes and the Restricted Elemental Runes under Crystalline Shimmer, so maybe breeding to a Ferrikki with Lucky Runes would heighten the chances of Crystalline Shimmer being visible on offspring).

There's also, in the Diamond Shop, in the Mutators, Dyes, and Disruptors section, Trait Extractors made specifically for these traits. You can use these trait extractors on a pet that has one of these restricted traits, to get a Scroll that contains the trait you took off another pet, and then put it on a pet that matches the criteria for using the Trait Scroll.

I did that to this pet, because I wanted her to have the restricted LG trait Mystical Unicorn Tail(The image is an image link leading to her profile):



For certain events, you can get exclusive pets that have these traits visible, such as the recent Patriotic Event where the themes you could obtain from the event(Patriotic and Blazing Fireworks), could have these traits on them(Some examples below):
https://www.sylestia.com/view/pets/?petid=7812857
https://www.sylestia.com/view/pets/?petid=7839410
https://www.sylestia.com/view/pets/?petid=7811161

During past Summer Fests, you could also get pets with these traits by collecting and combining Sea Shells(Which also gave the pet a certified name*Which is why the name is Purple*):
https://www.sylestia.com/view/pets/?petid=7283410

Any wild Sylesti you encounter in the Lost Grove could possibly have one of these traits as well, they're just encountered at random chance. If you have a Genetic Scanner, it'll tell you if the pet has one of these traits as well, or if you don't see it in a battle log notification, if you have a genetic scanner, you can check the genetics of the Sylesti by hitting the little scanner icon, which will let you see the Sylesti's genetics.
The Sylesti doesn't have to be themed either, any Sylesti encountered in the Lost Grove has a chance of having one of these traits.
These are a couple I've found that have LG-restricted traits(Either Visible, Carried, or Hidden):
https://www.sylestia.com/view/pets/?petid=7811151
https://www.sylestia.com/view/pets/?petid=6238028
https://www.sylestia.com/view/pets/?petid=6737792
https://www.sylestia.com/view/pets/?petid=6425177
https://www.sylestia.com/view/pets/?petid=6285714

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Sorry for the long explanation, I wanted to try to explain the best I could about these traits. ^^: