Muzaton took part in FIS races and national junior races from November 2005 at the age of 15, initially taking part in all disciplines. The first appearances in the European Cup followed in December 2008. His first victory in an FIS race was achieved in February 2010 in the slalom discipline . At the 2010 Junior World Championships in Megève , he won the gold medal in the Super-G. As a junior world champion, he was automatically eligible to compete in the World Cup finals in Garmisch-Partenkirchen ; in his World Cup debut on March 11, 2010, however, he did not finish.
In the course of the 2010/11 season, a specialization in the fast disciplines gradually began to emerge, but Muzaton could not classify himself in the points in several World Cup races. In contrast, he was in the top ten three times in the European Cup. Due to an injury, he had to pause almost the entire 2011/12 season. Muzaton won his first World Cup points on January 18, 2013 with 24th place in the super combination of Wengen . He achieved his best World Cup result so far on February 23, 2013 when he finished 13th in the downhill from Garmisch-Partenkirchen with the high starting number 50. In January 2017, he surprisingly achieved his first World Cup podium with second place in the combination of Wengen .
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