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Garde, who joined the Ligue 1 outfit in 2011, has led Lyon to two trophies during his tenure at the Stade de Gerland, winning the Coupe de France and Trophee des Champions in 2012.
Lyon sit fifth in the French top flight with one game to play, with Garde engineering an impressive season that has seen them reach the quarter-finals of the UEFA Europa League despite limited financial resources.
"Remi will not extend his contract despite multiple proposals on our side," Aulas said.
"'Remi took this decision based on personal and family reasons.
"We hold the man in high esteem - but Remi has also performed tremendously given the means he had been given.
"The profile of the new coach will have to fit with the project: invest in infrastructures and homegrown players.
"We'll try to find someone who fits, either because he's been with Lyon at some point, or because of his personality."
"I need to take a break and have some time off. This job is as harsh as it is amazing," Garde said.
"I want to thank the president for trusting me three years ago, when I was not a coach yet. We had a great collaboration together.
"I want to thank my players, especially those that believed in my work and the fact we could perform this year despite the conditions.
"I will not stay at the club, nor will I take another head coach job for now.
"It was no doubt a very hard decision for me. Helping all these young players to grow up was a great opportunity."
