A 4-dimensional Walker metrics with $c=0$ on a semi-Riemannian manifold $M$ have been investigated by E. Garc\'{\i}a-R\'{\i}o  and Y.Matsushita. The case $c=constant$ has been studied in [1]. In this paper we generalize these notions to the case of non--constant c. We find the form of the defining functions that makes this manifold similar to locally conformal flat 4-dimensional Walker manifold.