Integral constraints on the monodromy group of the hyperkähler resolution of a symmetric product of a K3 surface 1 Eyal Markman 2 Abstract: Let S [n] be the Hilbert scheme of length n subschemes of a K3 surface S. H 2 (S [n] , Z) is endowed with the Beauville-Bogomolov bilinear form. Denote by M on the subgroup of GL[H * (S [n] , Z)] generated by monodromy operators, and let M on 2 be its image in OH 2 (S [n] , Z). We prove that M on 2 is the subgroup generated by reflections with respect to +2 and −2 classes (Theorem 1.2). Thus M on 2 does not surject onto OH 2 (S [n] , Z)/(±1), when n − 1 is not a prime power.As a consequence, we get counter examples to a version of the weight 2 Torelli question for hyperkähler varieties X deformation equivalent to S [n] . The weight 2 Hodge structure on H 2 (X, Z) does not determine the bimeromorphic class of X, whenever n − 1 is not a prime power (the first case being n = 7). There are at least 2 ρ(n−1)−1 distinct bimeromorphic classes of X with a given generic weight 2 Hodge structure, where ρ(n − 1) is the Euler number of n − 1.The second main result states, that if a monodromy operator acts as the identity on H 2 (S [n] , Z), then it acts as the identity on H k (S [n] , Z), 0 ≤ k ≤ n + 2 (Theorem 1.5). We conclude the injectivity of the restriction homomorphism M on → M on 2 , if n ≡ 0, or n ≡ 1 modulo 4 (Corollary 1.6).