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We consider a modified Euler equation on $\mathbb R^2$. We prove existence ofweak global solutions for bounded (and fast decreasing at infinity) initialconditions and construct Gibbs-type measures on function spaces which arequasi-invariant for the Euler flow. Almost everywhere with respect to suchmeasures (and, in particular, for less regular initial conditions), the flow isshown to be also globally defined.

On a non-periodic modified Euler equation: existence and quasi-invariant measures