Petrov type-N solution for the null-surface formulation in $$2+1$$ 2 + 1 dimensions

A nontrivial solution of the field equations of the null-surface formulation in \(3+1\) dimensions has not, as yet, been found and, even in \(2+1\) dimensions, the field equations are extremely difficult to solve. Only two nontrivial solutions in \(2+1\) dimensions have previously been found and this work presents a third (exact) solution. It differs markedly from the two earlier solutions not only in its functional nature (parametric, as opposed to explicit or implicit) but also in its Petrov type and the physical nature of the source term. The solution is expressed in terms of elementary functions and has positive non-constant scalar curvature. The spacetime has a curvature singularity and the source of the gravitational field can be interpreted as a minimally coupled scalar field with a Liouville potential. The strong energy condition is violated but all of the other energy conditions hold. There is a congruence of null geodesic Killing vectors with the expansion, shear, and vorticity scalars all being zero.