The centrally symmetric V-states for active scalar equations. Two-dimensional Euler with cut-off

We consider the truncated equation for the centrally symmetric $V$-states in the 2d Euler dynamics of patches and prove the existence of solutions $y(x,λ),\, x\in [-1,1],\,λ\in (0,λ_0)$. These functions $y(x,λ)\in C^\infty(-1,1)$ and $\|y(x,λ)-|x|\|_{C[-1,1]}\to 0, λ\to 0$.

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