  Modified Derksen invariant HD^*(X) of an affine variety X is a subalgebra in
K[X] generated by kernels of all locally nilpotent derivations of K[X] with
slices. If there is a locally nilpotent derivation of K[X] with a slice then X
is a product of Y and a line, where Y is an affine variety. We prove that there
are three possibilities: A) HD^*(X) = K[X]; B) HD^*(X) is a proper infintely
generated subalgebra; C) HD^*(X) = \BK[Y]. We give examples for each case, and
also provide sufficient conditions for the variety Y so that the variety X
belongs to one of the type.
