  We continue our study of the local theory for quasiperiodic cocycles in
$\mathbb{T} ^{d} \times G$, where $G=SU(2)$, over a rotation satisfying a
Diophantine condition and satisfying a closeness-to-constants condition, by
proving a dichotomy between measurable reducibility (and therefore pure point
spectrum), and purely continuous spectrum in the space orthogonal to
$L^{2}(\mathbb{T} ^{d}) \hookrightarrow L^{2}(\mathbb{T} ^{d} \times G)$.
Subsequently, we describe the equivalence classes of cocycles under smooth
conjugacy, as a function of the parameters defining their K.A.M. normal form.
Finally, we derive a complete classification of the dynamics of one-frequency
($d=1$) cocycles over a Recurrent Diophantine rotation.
  All theorems will be stated sharply in terms of the number of frequencies
$d$, but in the proofs we will always assume $d=1$, for simplicity in
expression and notation.
