  We study the contact geometry of scalar second order hyperbolic equations in
the plane of generic type. Following a derivation of parametrized
contact-invariants to distinguish Monge-Ampere (class 6-6), Goursat (class 6-7)
and generic (class 7-7) hyperbolic equations, we use Cartan's equivalence
method to study the generic case. An intriguing feature of this class of
equations is that every generic hyperbolic equation admits at most a
nine-dimensional contact symmetry algebra. The nine-dimensional bound is sharp:
normal forms for the contact-equivalence classes of these maximally symmetric
generic hyperbolic equations are derived and explicit symmetry algebras are
presented. Moreover, these maximally symmetric equations are Darboux
integrable. An enumeration of several submaximally symmetric (eight and
seven-dimensional) generic hyperbolic structures is also given.
