Viviane, An interesting but complex question. Let me take a whack at it. The conditional "would" can refer to the present and the future. The perfect infinitive refers to something completed. I'd like to find. ...a present or future wish to find I'd like to have found.....a present regret about not finding it in the past (already) I would have liked to find......a past regret about not finding I would have liked to have found....a past regret about not having already found When I grow up I would like to find. future When I grow up I would like to have found......already found

Good question. Mind if I have a burl? "I would have liked to..." naturally looks back at the past, and imagines something different. "I'd like to..." looks forward. The fact that a perfect tense follows doesn't really confuse the grammar. Tuck it into future perfect, compare it and you'll see it works fine. (That is, if the writer was actually talking about a wish.)

subjunctive mood。 "I would have liked to find out what happened to X" is better