G(3)-supergeometry and a supersymmetric extension of the Hilbert–Cartan equation

Abstract We realize the simple Lie superalgebra G ( 3 ) as supersymmetry of various geometric structures, most importantly super-versions Hilbert–Cartan equation (SHC) and Cartan's involutive PDE system that exhibit 2 symmetry. provide symmetries explicitly compute, via first Spencer cohomology groups, Tanaka–Weisfeiler prolongation negatively graded superalgebras associated with two particular choices parabolics. discuss non-holonomic superdistributions growth vector | 4 , 1 0 deforming flat model SHC, prove second group gives a binary quadratic form, thereby indicating “square-root” classical quartic invariant for generic rank distributions in 5-dimensional space. Finally, we obtain super-extensions submaximally symmetric models, compute their observe dimension gap phenomenon.