The manifold $^*g-ESX_n$ is a generalized n-dimensional Riemannian manifold on which the differential geometric structure is imposed by the unified field tensor $^*g^{{lambda}{ u}}$ through the ES-connection which is both Einstein and semi-symmetric. The purpose of the present paper is to study the algebraic geometric structures of 3-dimensional $^*g-ESX_3$. Particularly, in 3-dimensional $^*g-ESX_3$, we derive a new set of powerful recurrence relations in the first class.