I would like to know the meaning of "Satanic Verses" and whether they exist.

The questioner is inclined towards the story cited by the exegetes in their interpretation of the verse: "And We did not send before you any messenger or prophet except that when he desired, Satan interjected into his desire" [Al-Hajj:52]. The gist of it is that the Prophet (peace be upon him) recited Surat An-Najm, and when he reached "Have you then considered Al-Lat and Al-'Uzza? And Manat, the third, the other?" [An-Najm:19-20], Satan interjected upon his tongue: "Those are the high-soaring cranes, and indeed their intercession is hoped for." Then the Prophet and the polytheists prostrated. However, the preservers of Hadith have deemed its chain of narration and its text weak. Ibn Khuzaymah said: "It is from the fabrication of the Zandaqah (heretics)," and Ibn Al-Arabi said: "It is false and has no basis." Qadi Iyad rejected it, stating that the Prophet (peace be upon him) is infallible from any addition or omission in revelation, and Satan has no power over him.

If it is assumed that the narration has some origin, then the meaning is that Satan added those words during the Prophet's recitation (peace be upon him) during the pauses in his reading, and the polytheists thought that he had uttered them.