  We make precise the structure of the first two reduction morphisms associated
with codimension two nonsingular subvarieties of quadrics $\Q{n}$, $n\geq 5$.
We give a coarse classification of the same class of subvarieties when they are
assumed to be not of log-general-type.
  Keywords: Adjunction Theory, classification, codimension two, conic bundles,
low codimension, non log-general-type, quadric, reduction, special variety
