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 prove this in the "only if" direction and provide a series of examples to support the converse one. We also conjecture that the unique ergodicity remains a generic property if the restrictions on the parameters do not contain a Lagrangian subspace.