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Real Madrid director general Jose Angel Sanchez used the case of head coach Zinedine Zidane's sons to argue the club have been wrongly sanctioned by FIFA.
Real and Atletico Madrid were found to have breached rules on the international transfer and registration of players aged under 18 and have been banned from registering new signings for the next two transfer windows.
The ban does not affect the current registration period and does not prevent players leaving either club but means that, after this month's transfer window closes, neither Real nor Atletico will be able to register any new players until the summer of 2017.
Atletico have also been hit with a 900,000 Swiss francs fine (?620,000) while Real have been fined 360,000 Swiss francs (?250,000).
Both clubs immediately announced their intention to appeal, with Sanchez and Real's director of legal services Javier Lopez Farre appearing at a press conference at the Bernabeu on Thursday evening.
A statement on the Real website laid out the club's response to the three rule breaches of which they have been accused, all of which they dispute.
Quoted in Marca, Sanchez said the punishment was "totally unexpected" and "absolutely unjustified", and revealed Zidane's sons were involved.
All four of the former France international's children are on Real's books.
Sanchez said: "The arguments given are so flawed that they've resulted in the club being sanctioned over the coach's sons.
"The case of the Zidane family or that of (former Real player Ezequiel) Garay's brother are well known and are an example of how absurd this is. I can assure you that the other cases are the same."
Sanchez continued: "It's completely untrue that we fielded players aged under 18 without registering them with the RFEF (Spanish Football Association) and that we failed to comply with the regulatory requirements.
"This club has a set of ethical values and the people in charge in this area have never violated them. The arguments made are erroneous."
Sanchez also said the club had "high hopes" of success through FIFA's appeals procedure but would go to the Court of Arbitration for Sport (CAS) if necessary.
Thursday's announcement comes after FIFA punished Barcelona with a year-long ban in April 2014 for the transfer of players under the age of 18.
Barca appealed to both FIFA and CAS against the ruling but ultimately lost both bids. However, with the ban being suspended while the appeals were ongoing, Barca were able to sign six players, including Luis Suarez, in the summer of 2014.
The Catalan club eventually served their punishment during the two transfer windows in 2015, but they were still able to sign Arda Turan and Aleix Vidal from Atletico Madrid and Sevilla respectively for a combined fee of ?37million last summer.
Barca were not able to register Turan and Vidal as players until this month, though, and the duo have only recently been able to make their debuts for the club.
A FIFA spokesperson confirmed to Press Association Sport that Real and Atletico would also be able to sign new players during their bans, but would not be able to register them.