  Remarkable anisotropic structures have been recently observed in the order
parameter Delta_k of the underdoped superconductor Bi-2212. Such findings are
strongly suggestive of deviations from a simple d_{x^2 - y^2}-wave picture of
high-Tc superconductivity, i.e. Delta_k ~ cos (k_x) - cos (k_y). In particular,
flatter nodes in Delta_k are observed along the k_x = (+/-) k_y directions in
k-space, than within this simple model for a d-wave gap. We argue that
nonlinear corrections in the k-dependence of Delta_k near the nodes introduce
new energy scales, which would lead to deviations in the predicted power-law
asymptotic behaviour of several measurable quantities, at low or intermediate
temperatures. We evaluate such deviations, either analytically or numerically,
within the interlayer pair-tunneling model, and within yet another
phenomenological model for a d-wave order parameter. We find that such
deviations are expected to be of different sign in the two cases. Moreover, the
doping dependence of the flatness of the gap near the nodes is also
attributable to Fermi surface effects, in addition to possible screening
effects modifying the in-plane pairing kernel, as recently proposed.
