Laguerre isoparametric hypersurfaces in .n with two distinct non-zero principal curvatures

An umbilical free oriented hypersurface x: M . .n with non-zero principal curvatures is called a Laguerre isoparametric hypersurface if its Laguerre form C=.iCi.i=.i..1(Ei(log.)(r.ri).Ei(r)).i vanishes and Laguerre shape operator S=..1(S.1.rid) has constant eigenvalues. Here . = . i (r . r i )2, r=r1+r2+.+rn.1n.1 is the mean curvature radius and S is the shape operator of x. {E i } is a local basis for Laguerre metric g = . 2III with dual basis {. i } and III is the third fundamental form of x. In this paper, we classify all Laguerre isoparametric hypersurfaces in .n(n > 3) with two distinct non-zero principal curvatures up to Laguerre transformations.