Learn Finnish: Sinnikäs tutkija uskoo löytävänsä uuden vision. - The persistent researcher believes she will find a new vision.

This sentence uses the referative construction (partisiippirakenne) to replace an että (that) clause. Because the subject of the main clause (tutkija) is the exact same person doing the finding, Finnish drops the pronoun hänen. Instead, the 3rd person possessive suffix -nsä is attached directly to the participle to show "she herself will find".

If the subjects were different (e.g., she believes someone else will find it), how would this sentence change?

If she believed someone else would find it, you wouldn't use the possessive suffix on the participle. You would use the genitive pronoun for the other person and the normal genitive form of the participle: uskoo hänen löytävän (she believes that he/she will find).

The referative construction usually requires the participle to be in the genitive case (like löytävän). Where did the genitive -n go in löytävänsä?

It dropped out! A standard rule in Finnish is that when you add a possessive suffix (like -nsä) to a word ending in a single genitive -n, the -n disappears. So, löytävän + -nsä becomes löytävänsä.

The English translation says "will find", which is future tense. Why does löytävänsä use the present participle (-vä)?