  We propose the Legendrian web in a contact three manifold as a second order
generalization of the planar web. An Abelian relation for a Legendrian web is
analogously defined as an additive equation among the first integrals of its
foliations. For a class of Legendrian $\, d$-webs defined by simple second
order ODE's, we give an algebraic construction of $$ \rho_d =
\frac{(d-1)(d-2)(2d+3)}{6}$$ linearly independent Abelian relations. We then
employ the method of local differential analysis and the theory of linear
differential systems to show that $\, \rho_d$ is the maximum rank of a
Legendrian $\, d$-web.
  In the complex analytic category, we give a possible projective geometric
interpretation of $\rho_d$ as an analogue of Castelnuovo bound for degree 2d
surfaces in the 3-quadric $\mathbb{Q}^3\subset\mathbb{P}^4$ via the duality
between $\mathbb{P}^3$ and $\mathbb{Q}^3$ associated with the rank two complex
simple Lie group Sp$(2,\mathbb{C})$.
  The Legendrian 3-webs of maximum rank three are analytically characterized,
and their explicit local normal forms are found. For an application, we give an
alternative characterization of a Darboux super-integrable metric as a two
dimensional Riemannian metric $\,g_+$ which admits a mate metric $\, g_-$ such
that a Legendrian 3-web naturally associated with the geodesic foliations of
the pair $\, g_{\pm}$ has maximum rank.
