Alicorn ponies can be naturally born in any family, though they are very rare. They can’t do everything a real alicorn can do, however. They either can’t use their magic or they can’t fly. That is because of the genetic defects. In such alicorn’s DNA it is written that they are either a pegasus or a unicorn despite them having both wings and a horn.
There are cases, however, when an alicorn is able to use both wings and the horn. It is a very, very rare case though, that happens once in a few centuries.
