The geometry and invariance properties for certain classes of metrics with neutral signature

In this paper, we study anti-self dual manifolds endowed with metrics of neutral signature. Since the metrics depend on solutions of, in some cases, well-known partial differential equations (PDEs), we determine exact solutions using Lie group methods. This concludes specific forms of the metrics. We then determine the isometries and the variational symmetries of the underlying metrics and corresponding Euler-Lagrange (geodesic) equations, respectively, and establish relationships between the resultant Lie algebras. In some cases, we construct conservation laws via these symmetries or the "multiplier approach".