When a wizarding monarch dies, the order of succession is as follows: firstly, their named heir; secondly, the monarch’s oldest eligible descendent; and finally oldest eligible member of the royal family. Eligibility requirements are one: being of age (17 or older), two: not being a past monarch who lost the throne in a challenge, and three: being still alive (ghosts and portraits cannot be the current monarch)
When Bellatrix and Voldemort died, their daughter, Delphi, was too young, and Pollux had already been king and lost the throne in a challenge making him ineligible. As a result, the line of succession passed to Pegasus, who was the oldest royal remaining after Pollux. Her younger, twin brother, Cygnus believed that squibs were unable to inherit the throne and that therefore he was the rightful king. At this time, a majority of wizarding Great Britain was unaware that Pegasus and Hugo were even alive and believed Cygnus was the rightful monarch.
The Order of the Phoenix was not thrilled with the sort of king Cygnus was shaping up to be. He pardoned several Death Eaters, including Dolores Umbridge, his school-hood friend, Lucius Malfoy, his son-in-law, and Corban Yaxley, his nephew by marriage. He also appointed Yaxley one of the temporary Ministers of Magic, the other two being Kingsley Shacklebolt, who was appointed by the Order and Cornelius Fudge who was appointed by the Wizengamot. After the battle at the Ministry, Fudge technically quit before he was forced to resign, so he was still eligible to become Minister again. Cygnus wanted Yaxley to be the named the full Minister of Magic.
Lucius tried to convince Cygnus, that with the Black family seemingly dying out, a peaceful transfer of power to a new royal family, say the Malfoys, would be in the best interest of the wizarding world. Cygnus seeing the threat to his power behind Lucius’ ideas, convinced his daughter, Andromeda to kill him. He feigned a lack of involvement in the plot after Lucius’ death.
Amidst the chaos, Pegasus and Hugo reappeared, stating her claim to throne. The Orange Pips, the group Pegasus, Hugo, and Alphard had formed durning the first wizarding war to oppose the Death Eaters, unanimously supported them. They were later joined by the Order of the Phoenix and Dumbledore’s Army. Eventually, Cygnus was forced to recede his claim and acknowledge Pegasus and Hugo as the rightful Queen and King. Pegasus and Hugo named their middle son Florizel and this wife Jean as their heirs.
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