Spectral anomaly for interacting fermions is characterized by the spectral function A ([k − k F ], ω) satisfying the scaling relation A (Λy1 [k − k F ], Λy2 ω) =ΛyA A ([k − k F ], ω), where y1, y2, and yA are the exponents defining the universality class. For a Fermi liquid y1 = 1, y2 = 1, yA = −1; all other values of the exponents are termed anomalous. In this paper, an example for which y1 = 1, y2 = 1, but yA = α − 1 is considered in detail. Attractive interaction added to such a critical system leads to a novel superconducting state, which is explored and its relevance to high temperature cuprate superconductors is discussed.