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We study explicit solutions to the 2 dimensional Euler equations in theLagrangian framework. All known solutions have been of the separation ofvariables type, where time and space dependence are treated separately. Thefirst such solutions were known already in the 19th century. We show that allthe solutions known previously belong to two families of solutions andintroduce three new families of solutions. It seems likely that these are allthe solutions that are of the separation of variables type.

On the explicit solutions of separation of variables type for the incompressible 2D Euler equations