A Lorentzian hypersurface x : M^4_1 → L^5, isometrically immersed into the Lorentz-Minkowski 5-space L^5 , is said to be L1-biconservative if the tangent component of vector field (L_1)^2 x is identically zero, where L_1 is the linearized operator associated to the first variation of 2nd mean curvature vector field on M_1^4. Since L0 = ∆ is the well known Laplace operator, the concept of L1-biconservative hypersurface is an extension of ordinary conservativity. The biconservativity is related to the physical concept of conservative stress-energy with respect to the bienergy functional. We discuss on Lorentzian hypersurfaces of L^5 having at most two distinct principal curvatures. After illustrating some examples, we prove that every L1-bicoservative Lorentzian hypersurface with constant ordinary mean curvature and at most two distinct principal curvatures in L^5 has to be of constant 2nd mean curvature.
