Reduced Tangent Cones and Conductor at Multiplanar Isolated Singularities

We prove a result on the associated graded ring of local algebras that satisfy some appropriate conditions. This allows us to compute, in the general case, the conductor of the local ring of singular isolated points on surfaces whose tangent cone is multiplanar; that is, it consists - as a set - of planes. We explicitly construct a class of such surfaces in the rational case.