  Given a (transitive or non-transitive) Anosov vector field $X$ on a closed
three-dimensional manifold $M$, one may try to decompose $(M,X)$ by cutting $M$
along two-tori transverse to $X$. We prove that one can find a finite
collection $\{T_1,\dots,T_n\}$ of pairwise disjoint, pairwise non-parallel
incompressible tori transverse to $X$, such that the maximal invariant sets
$\Lambda_1,\dots,\Lambda_m$ of the connected components $V_1,\dots,V_m$ of
$M-(T_1\cup\dots\cup T_n)$ satisfy the following properties: 1, each
$\Lambda_i$ is a compact invariant locally maximal transitive set for $X$, 2,
the collection $\{\Lambda_1,\dots,\Lambda_m\}$ is canonically attached to the
pair $(M,X)$ (i.e., it can be defined independently of the collection of tori
$\{T_1,\dots,T_n\}$), 3, the $\Lambda_i$'s are the smallest possible: for
every (possibly infinite) collection $\{S_i\}_{i\in I}$ of tori transverse to
$X$, the $\Lambda_i$'s are contained in the maximal invariant set of $M-\cup_i
S_i$. To a certain extent, the sets $\Lambda_1,\dots,\Lambda_m$ are analogs
(for Anosov vector field in dimension 3) of the basic pieces which appear in
the spectral decomposition of a non-transitive axiom A vector field. Then we
discuss the uniqueness of such a decomposition: we prove that the pieces of the
decomposition $V_1,\dots,V_m$, equipped with the restriction of the Anosov
vector field $X$, are "almost unique up to topological equivalence".
