Affine-ruled varieties without the Laurent cancellation property

We describe a method to construct hypersurfaces of the complex affine n-space with isomorphic C*-cylinders. Among these hypersurfaces, we find new explicit counterexamples to the Laurent cancellation problem, that is, hypersurfaces that are nonisomorphic, although their C*-cylinders are isomorphic as abstract algebraic varieties. We also provide examples of nonisomorphic varieties X and Y with isomorphic Cartesian squares X x X and Y x Y.