  The paper is devoted to the complete classification of all real Lie algebras
of contact vector fields on the first jet space of one-dimensional submanifolds
in the plane. This completes Sophus Lie's classification of all possible Lie
algebras of contact symmetries for ordinary differential equations. As a main
tool we use the abstract theory of filtered and graded Lie algebras. We also
describe all differential and integral invariants of new Lie algebras found in
the paper and discuss the infinite-dimensional case.
