Classifications of Kantowski-Sachs, Bianchi types I and III spacetimes according to Ricci collineations

The Ricci collineation classifications of Kantowski-Sachs, Bianchi types I and III space-times are studied according to their degenerate and non-degenerate Ricci tensor. When the Ricci tensor is degenerate, the special cases are classified and it is shown that there are many cases of Ricci collineations (RCs) with infinite number of degrees of freedom, and the group of RCs is ten-dimensional in some spacial cases. Furthermore, it is found that when the Ricci tensor is non-degenerate, the group of RCs is finite-dimensional, and we have only either four which coincides with the isometries or six proper RCs in addition to the four isometries.