  In the scalar 1D case, conservation laws and Hamilton-Jacobi equations are
deeply related. For both, we characterize those profiles that can be attained
as solutions at a given positive time corresponding to at least one initial
datum. Then, for each of the two equations, we precisely identify all those
initial data yielding a solution that coincide with a given profile at that
positive time. Various topological and geometrical properties of the set of
these initial data are then proved. 2000 Mathematics Subject Classification:
35L65, 35F21, 93B30, 35R30.
