Determinacy of smooth germs with real isolated line singularities

The germ of a smooth real-valued function on Euclidean space is called a re al isolated line singularity if its singular set is a nonsingular curve, it s Jacobian ideal is Lojasiewicz at the singular set, and its Hessian determ inant restricted to the singular set is Lojasiewicz at 0. Consider the set of all germs whose singular set contains a fixed nonsingular curve L. We pr ove that such a germ f is infinitely determined among all such germs with r espect to composition by diffeomorphisms preserving L if, and only if, the Jacobian ideal of f contains all germs which vanish on L and are infinitely at at 0 if, and only if, f is a real isolated line singularity.