A parametrization of the theta divisor of the quartic double solid

Let M(2;0,3) be the moduli space of rank-2 stable vector bundles with Chern classes c_1=0, c_2=3 on the Fano threefold X, the double solid of index two. We prove that the vector bundles obtained by Serre's construction from smooth elliptic quintic curves on X form an open part of an irreducible component M' of M(2;0,3) and that the Abel-Jacobi map F:M'-->J(X) into the intermediate Jacobian J(X) defined by the second Chern class is generically finite of degree 84 onto a translate of the theta divisor. We also prove that the family of elliptic quintics on a general X is irreducible and of dimension 10.