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Applying an original megaideal-based version of the algebraic method, wecompute the point-symmetry pseudogroup of the dispersionless (potentialsymmetric) Nizhnik equation. This is the first example of this kind in theliterature, where there is no need to use the direct method for completing thecomputation. The analogous studies are also carried out for the correspondingnonlinear Lax representation and the dispersionless counterpart of thesymmetric Nizhnik system. We also first apply the megaideal-based version ofthe algebraic method to find the contact-symmetry (pseudo)group of a partialdifferential equation. It is shown that the contact-symmetry pseudogroup of thedispersionless Nizhnik equation coincides with the first prolongation of itspoint-symmetry pseudogroup. We check whether the subalgebras of the maximal Lieinvariance algebra of the dispersionless Nizhnik equation that naturally arisein the course of the above computations define the diffeomorphisms stabilizingthis algebra or its first prolongation. In addition, we construct all thethird-order partial differential equations in three independent variables thatadmit the same Lie invariance algebra. We also find a set of geometricproperties of the dispersionless Nizhnik equation that exhaustively defines it.

Point- and contact-symmetry pseudogroups of dispersionless Nizhnik equation