  Let $P$ be a pointed, closed convex cone in $\mathbb{R}^d$. We prove that for
two pure isometric representations $V^{(1)}$ and $V^{(2)}$ of $P$, the
associated CAR flows $\beta^{V^{(1)}}$ and $\beta^{V^{(2)}}$ are cocycle
conjugate if and only if $V^{(1)}$ and $V^{(2)}$ are unitarily equivalent. We
also give a complete description of pure isometric representations of $P$ with
commuting range projections that give rise to type I CAR flows. We show that
such an isometric representation is completely reducible with each irreducible
component being a pullback of the shift semigroup $\{S_t\}_{t \geq 0}$ on
$L^2[0,\infty)$. We also compute the index and the gauge group of the
associated CAR flows and show that the action of the gauge group on the set of
normalised units need not be transitive.
