In Grebenev, Wac\l{}awczyk, Oberlack (2019 Phys A: Math. Theor. 52, 33) the conformal invariance (CI) of the characteristic ${\bf X}_{1}(t)$ (the zero-vorticity Lagrangian path) of the first equation (i.e. for the evolution of the $1$-point PDF $f_1({\bf x}_{1},\omega_{1},t)$, ${\bf x}_{1} \in D_1 \subset \Bbb R^2)$ of the inviscid Lundgren-Monin-Novikov (LMN) equations for $2d$ vorticity fields was derived. The infinitesimal operator admitted by the characteristics equation generates an infinite dimensional Lie pseudo-group $G$ which conformally acts on $D_1$. We define the conformal invariant differential form $ds^2 =  f_1\cdot\left({dX_{1}^{\it 1}}^2 + {dX_{1}^{\it 2}}^2\right)$ along the characteristic $\left.{\bf X}_{1}(t)\right|_{\omega_{1} = 0}$ together with the simple action functional ${\mathcal F}({\bf X}_{1},ds^2)$. We demonstrate that $G_{\mathcal Y}$, which is a subgroup of the group $G$ restricted on the variables ${\bf x}_{1}$ and $f_1$, gives rise to a symmetry transformations of  ${\mathcal F}({\bf X}_{1},ds^2)$. With this, we calculate the second-order universal differential invariant $J_2^{\mathcal Y}$ (or the multiscale representation of the invariants) of $G_{\mathcal Y}$ under the action on the zero-vorticity characterisctics. We show that ${\mathcal F}({\bf X}_{1},ds^2)$ is a scalar invariant and generates all differential invariants, which look like the quantities of different scales, from $J_2^{\mathcal Y}$  by the operators of invariant differentiation. It gives insight into the geometry of a flow domain nearby point ${\bf x}_{1}$ in the sense of Cartan.