We study the scale-covariant theory of gravitation proposed by Canuto under the self-similar hypothesis.By considering the matter collineation approach,we deduce the form of the main quantities and obtain some restrictions on the behavior of the gauge function,φ(t).We apply the obtained results to study the Bianchi I,Ⅶo,Ⅸ and Kantowsky Sachs models,and arrive at the conclusion that instead of considering a varying G theory,the solution is only valid for a particular value of the parameter of the equation of state,γ,.This leads us to obtain φ(t) =const,and by taking into account the exteal condition,Gφ2 =const,we may arrive at the conclusion that G=const.Therefore,all the obtained solutions are the same as the ones obtained in the framework of general relativity.