  It is known since 40 years old paper by M. Keane that minimality is a generic
(i.e. holding with probability one) property of an irreducible interval
exchange transformation. If one puts some integral linear restrictions on the
parameters of the interval exchange transformation, then minimality may become
an "exotic" property. We conjecture in this paper that this occurs if and only
if the linear restrictions contain a Lagrangian subspace of the first homology
of the suspension surface. We partially prove it in the "only if" direction and
provide a series of examples to support the converse one. We show that the
unique ergodicity remains a generic property if the restrictions on the
parameters do not contain a Lagrangian subspace (this result is due to Barak
Weiss).
