  Coalescents with multiple collisions (also called Lambda-coalescents or
simple exchangeable coalescents) are used as models of genealogies. We study a
new class of Markovian coalescent processes connected to a population model
with immigration. Imagine an infinite population with immigration labelled at
each generation by N:={1,2,...}. Some ancestral lineages cannot be followed
backwards after some time because their ancestor is outside the population. The
individuals with an immigrant ancestor constitute a distinguished family and we
define exchangeable distinguished coalescent processes as a model for genealogy
with immigration, focussing on simple distinguished coalescents, i.e such that
when a coagulation occurs all the blocks involved merge as a single block.
These processes are characterized by two finite measures on [0,1] denoted by
M=(\Lambda_{0},\Lambda_{1}). We call them M-coalescents. We show by martingale
arguments that the condition of coming down from infinity for the M-coalescent
coincides with that obtained by Schweinsberg for the \Lambda-coalescent. In the
same vein as Bertoin and Le Gall, M-coalescents are associated with some
stochastic flows. The superprocess embedded can be viewed as a generalized
Fleming-Viot process with immigration. The measures \Lambda_{0} and \Lambda_{1}
specify respectively the reproduction and the immigration. The coming down from
infinity of the M-coalescent will be interpreted as the initial types
extinction: after a certain time, all individuals are immigrant children.
