On role of symmetries in Kelvin wave turbulence

E.V. Kozik and B.V. Svistunov (KS) paper Symmetries and Interaction Coefficients of Kelvin waves, arXiv:1006.1789v1, [cond-mat.other] 9 Jun 2010, contains a comment on paper Symmetries and Interaction coefficients of Kelvin waves, V. V. Lebedev and V. S. Lvov, arXiv:1005.4575, 25 May 2010. It relies mainly on the KS text Geometric Symmetries in Superfluid Vortex Dynamics}, arXiv:1006.0506v1 [cond-mat.other] 2 Jun 2010. The main claim of KS is that a symmetry argument prevents linear in wavenumber infrared asymptotics of the interaction vertex and thereby implies locality of the Kelvin wave spectrum previously obtained by these authors. In the present note we reply to their arguments. We conclude that there is neither proof of locality nor any refutation of the possibility of linear asymptotic behavior of interaction vertices in the texts of KS.