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It was discovered experimentally in (Bernard et al. Nature Physics, 2006) that the zero-vorticity isolines in 2D turbulence belong to the class of conformal invariant $SLE_\kappa$ (Schram-Lowner evolution) curves with $\kappa = 6$. With this motivation, we performed a Lie group analysis in of the first equation (i.e. for the evolution of the 1-point probability density function (PDF) $f_1(x_{(1)},\omega_{(1)},t)$) of the inviscid Lundgren-Monin-Novikov (LMN) equations for 2D vorticity fields. We proved that the conformal group (CG) is broken for the 1-point PDF but the CG is recovered for the equation restricted on the characteristics with zero-vorticity. The main focus of the present work is directed to a Lie group analysis of the characteristic equations of the inviscid LMN hierarchy truncated to the first equation. Besides the derivation of the CG invariance of the zero-vorticity isolines, we demonstrate that the infinitesimal operator admitted by the characteristic equations forms a Lie algebra which is the Witt algebra, whose central extension represents exactly the Virasoro algebra. This is a joint work with M. Waclawczyk (University of Warsaw, Poland) and M. Oberlack (TU Darmstadt, Germany). The results are published in J. Phys. A: Math. Theor. 2017. 50(43) P. 435502-44.