Hello Kubarola,
Thanks for that info and for choosing the Apple Support Communities. If we understand correctly, you are unable to find Live Photos you took while on a FaceTime call on your iPhone. First, let's make sure your iPhone is fully up to date Update your iPhone or iPad - Apple Support as there can be fixes for issues like yours when you update.
If you are fully up to date already, try temporarily turning off FaceTime in Settings > FaceTime, then force close the Settings app (Close an app on your iPhone or iPod touch - Apple Support) and restart your iPhone: Restart your iPhone - Apple Support
Then go back to Settings > FaceTime and turn FaceTime back on, make sure your Apple ID is shown for "You can be reached by FaceTime at" and then test this issue with Live Photos during a FaceTime call again: Take a Live Photo in FaceTime on iPhone - Apple Support
Cheers!