  Spectral anomaly for interacting Fermions is characterized by the spectral
function $A([k-k_F],\omega)$ satisfying the scaling relation $A(\Lambda^{y_1}
[k-k_F],\Lambda^{y_2}\omega)= \Lambda^{y_A}A([k-k_F],\omega)$, where $y_1$,
$y_2$, and $y_A$ are the exponents defining the universality class. For a Fermi
liquid $y_1=1$, $y_2=1$, $y_A=-1$; all other values of the exponents are termed
anomalous. In this paper, an example for which $y_1=1$, $y_2=1$, but
$y_A=\alpha-1$ 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.
