  Attractiveness is a fundamental tool to study interacting particle systems
and the basic coupling construction is a usual route to prove this property, as
for instance in simple exclusion. The derived Markovian coupled process
$(\xi_t,\zeta_t)_{t\geq 0}$ satisfies: (A) if $\xi_0\leq\zeta_0$
(coordinate-wise), then for all $t\geq 0$, $\xi_t\leq\zeta_t$ a.s. In this
paper, we consider generalized misanthrope models which are conservative
particle systems on $\Z^d$ such that, in each transition, $k$ particles may
jump from a site $x$ to another site $y$, with $k\geq 1$. These models include
simple exclusion for which $k=1$, but, beyond that value, the basic coupling
construction is not possible and a more refined one is required. We give
necessary and sufficient conditions on the rates to insure attractiveness; we
construct a Markovian coupled process which both satisfies (A) and makes
discrepancies between its two marginals non-increasing. We determine the
extremal invariant and translation invariant probability measures under general
irreducibility conditions. We apply our results to examples including a
two-species asymmetric exclusion process with charge conservation (for which
$k\le 2$) which arises from a Solid-on-Solid interface dynamics, and a stick
process (for which $k$ is unbounded) in correspondence with a generalized
discrete Hammersley-Aldous-Diaconis model. We derive the hydrodynamic limit of
these two one-dimensional models.
