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Remarkable anisotropic structures have been recently observed in the orderparameter Delta_k of the underdoped superconductor Bi-2212. Such findings arestrongly suggestive of deviations from a simple d_{x^2 - y^2}-wave picture ofhigh-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 ink-space, than within this simple model for a d-wave gap. We argue thatnonlinear corrections in the k-dependence of Delta_k near the nodes introducenew energy scales, which would lead to deviations in the predicted power-lawasymptotic behaviour of several measurable quantities, at low or intermediatetemperatures. We evaluate such deviations, either analytically or numerically,within the interlayer pair-tunneling model, and within yet anotherphenomenological model for a d-wave order parameter. We find that suchdeviations are expected to be of different sign in the two cases. Moreover, thedoping dependence of the flatness of the gap near the nodes is alsoattributable to Fermi surface effects, in addition to possible screeningeffects modifying the in-plane pairing kernel, as recently proposed.Comment: 7 pages, 4 embedded PostScript figures. Uses svjour, epsfig, amsmath,  amssymb, xspace. To be published in Eur. Phys. J.

Extended -wave superconductivity. Flat nodes in the gap and the low-temperature asymptotic properties of high- superconductors