Isoparametric hypersurfaces of Lorentz-Minkowski spaces, which has been classified by M.A. Magid in 1985, have motivated some researchers to study biconservative hypersurfaces. A biconservative hypersurface has conservative stress-energy with respect to the bienergy functional. A timelike (Lorentzian) hypersurface x : M^n_1 → E^{n+1}_1, isometrically immersed into the Lorentz-Minkowski space E^{n+1}_1, is said to be biconservative if the tangent component of vector field (∆^2)x on M^n_1 is identically zero. In this paper, we study the Lk-extension of biconservativity condition. The map L_k on a hypersurface (as the kth extension of Laplace operator L_0 = ∆) is the linearized operator arisen from the first variation of (k + 1)th mean curvature of hypersurface. After illustrating some examples, we prove that an L_k-biconservative timlike hypersurface of E^{n+1}_1, with at most two distinct principal curvatures and some additional conditions, is isoparametric.
