"The UEFA Executive Committee has warned both football associations that - irrespective of any decisions taken by the independent disciplinary bodies relating to incidents inside the stadium - it will not hesitate to impose additional sanctions ...including the potential disqualification of their respective teams from the tournament, should such violence occur again," UEFA said in a statement.
The French government subsequently announced that they are banning alcohol from cities hosting Euro 2016 matches and associated fan zones.
Police chiefs in nine hosting cities have been asked to take all measures to prevent the sale, consumption and transportation of alcohol on the day before a fixture and on the match day.
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France's interior minister Bernard Cazeneuve said, "It is absolutely necessary that the national federations whose supporters create incidents of this nature be punished for what happened inside the stadium, and also outside."
UEFA had earlier launched disciplinary proceedings against Russia in response to their fans storming their English counterparts in the stadium.
"We urge both the Football Association and the Russian Football Union to appeal to their supporters to behave in a responsible and respectful manner," UEFA said.
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UEFA opened the disciplinary proceedings against the Football Union of Russia after violence broke out at Saturday's game against England in Marseille, bringing charges against Russia for crowd disturbances, setting off fireworks and tracis behaviour.
UEFA added that it had not taken any action against the English FA for the skirmishes that ensued.
Trouble was thought to have begun towards the end of the match when flares were let off by Russian fans at the Stade Velodrome. Subsequent footage appeared to show Russian fans charging England supporters after the 1-1 draw between England and Russia.
Despite having previously downplayed the violence, Russian sports minister Vitaly Mutko says UEFA's decision to open a disciplinary case is the right decision, accepting that Russia had "behaved incorrectly," but added that there were "many nuances" to what happened.
Earlier on Sunday, Mutko was quoted as having told Russia Today: "There was no clash...that's being exaggerated, in fact everything is fine here," adding that organisation - including segregation between English and Russian fans - had been insufficient.
In a strongly-worded statement UEFA expressed "utter disgust" and "serious concern" over the scenes that unfolded, but acknowledged concerns about security in the stadium saying it would implement "corrective measures" to address these.
Meanwhile, a spokesman for France's Interior Ministry was quick to defend police in the face of those security concerns and some criticism of their handling of the violence.
Speaking on behalf of the French Government Pierre-Henri Brandet said: "If there is a failure, it is that of soccer which is sick because some of its fans drink excessively and fight."
Violence between fans in Marseille started several days ago with bloody scenes in the Old Port area, meanwhile unrest was also said to have broken out in Nice between locals and Northern Ireland fans ahead of their side's opening match against Poland on Sunday.
A total of 15 arrests had been made by Sunday afternoon, authorities in Marseille said while 35 people had been injured in the clashes.
UEFA said a decision on sanctions would be made within the next few days, once Russia had submitted written statements.
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