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 $y(x,\lambda),\, x\in [-1,1],\,\lambda\in (0,\lambda_0)$. These functions $y(x,\lambda)\in C^\infty(-1,1)$ and $\|y(x,\lambda)-|x|\|_{C[-1,1]}\to 0, \lambda\to 0$.