Abstract
The isovector fields (symmetry groups) of the Boyer-Finley equation of self-dual Einstein spaces of Euclidean signature with one rotational Killing vector are calculated using the geometric prolongation technique. Using the explicit determination of the component of the isovector field introduced by Suhubi (1991 Int. J. Eng. Sci. 29 133) several times to find appropriate similarity variables, we reduce the partial differential equation to ordinary differential equation. Also, we obtained many new exact invariant (similarity) solutions of the Boyer-Finley equation which generate solutions of Einstein fields equations.