It was clearly validated experimentally in Bernard et al (2006 Nat. Phys. 2 124–8) that the zero-vorticity isolines in 2D turbulence belong to the class of conformal invariant (Schram–Löwner evolution) curves with . The diffusion coefficient classifies the conformally invariant random curves. With this motivation, we performed a Lie group analysis in Grebenev et al (2017 Phys. A: Math. Theor. 50 5502–44) of the first of the inviscid Lundgren–Monin–Novikov (LMN) equations for 2D vorticity fields. This equation describes the evolution of the 1-point probability density function (PDF) . We proved that the conformal group (CG) is not admitted by the 1-point PDF equation itself, however it is permitted under the condition . The main focus of the present work is to prove explicitly the CG invariance of the zero-vorticity Lagrangian path, which is the characteristic of the inviscid LMN hierarchy truncated to the first equation. We also show the CG invariance of the separation and coincidence properties of the PDFs. Besides the derivation of the CG invariance of the zero-vorticity Lagrangian path, 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. The numerical value of the central charge c occurring here could not be calculated. Other mathematical tools need to be involved to link this analytical study with the previous analyses by Bernard et al, who report the value c  =  0, which corresponds to for the .