I came across your Itúrbide web page and I noticed that you mentioned my great-grand uncle Nicolás Fernández de Piérola Villena as a descendant of Emperor Agustín, through his first son and an illegitimate son in two of your genealogical reports, with the other saying he died childless.
I have in my possession the most authoritative biography of him, "Don Nicolás de Piérola" by Alberto Ulloa Sotomayor (2nd Edition, 1981, Lima) and it contains information that contradicts what you have.
Don Nicolás was born on January 5th, 1839 in Camaná, Arequipa, Perú and died on June 23rd, 1913 in Lima, Perú. I shall now translate pp. 96 and 97 of the book regarding him and the Itúrbide family.
The Imperial Legend
The wedding and death certificates of Doña Jesús state that she was the legitimate daughter of Don Joaquín de Itúrbide and Doña María del Carmen Villena y Pérez. According to many of Piérola's biographers, Don Joaquín (b. Madrid), was the son of Agustín de Itúrbide, the brave, brief and ambitious Emperor of Mexico in 1822, and of Doña Antonia Cadorna. The Emperor had 9 legitimate siblings, 7 of which had already been born by the time of the coronation, and thus were expressly mentioned in the decree of the congress of Mexico of June 22nd, 1822. None was called Joaquín, this name belonging only to José Joaquín de Itúrbide y Arregui, the Emperor's father. According to my investigations, neither him [the Emperor] nor the majority of his siblings ever came to Peru, save the firstborn, Agustín Gerónimo de Itúrbide y Huarte, who served under Bolívar in the Battle of Ayacucho [N.B. December 9th, 1824]. Since he went back to Colombia and then to Mexico after Bolívar's death [N.B. December 17th, 1830], he could not have had a daughter born here [Peru] in 1841; neither could he have been the husband of María del Carmen Villena, because he never married.
The Emperor had no illegitimate siblings. Don Joaquín, the father of Doña Teresa Itúrbide Villena, Piérola's wife, was never in Mexico nor was his alleged father ever in Peru. He was not a member of the imperial family nor of any other family , whose surname still exists in Mexico.
Who was he, then? Perhaps an adventurer who in order to gain prestige during the vibrant independence times, made himself a son of Augustín I, to whom the consolidation of the Mexican independence is owed and the romantic legend of an imperial adventure. However he was born in Madrid and the Emperor was never in Spain. He might have had a long lost (and distant) relative far away in Navarre in the early 18th century, when the imperial Itúrbides left for New Spain. But Doña Jesús de Itúrbide y Villena, Piérola's wife, was not the granddaughter of Agustín I; and surely she and Don Nicolás knew it, due to the fact that no-one ever heard them claim such lineage. In spite of this, the Caudillo [N.B. one of Don Nicolás' nicknames, meaning Leader], knowing full well that the distant imperial shine helped to colour his own legend, let the rumour be repeated and published.
The author quotes two sources:
a) Romero de Terrero, Manuel: La Corte de Agustín I, Emperador de México. Mexico 1922.
b) Mestas, Alberto: Agustín de Itúrbide, Emperador de Méjico. San Sebastián, 1939.
I hope this information will be useful.