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Three new examples of 4-dimensional irreducible symplectic V-manifolds areconstructed. Two of them are relative compactified Prymians of a family ofgenus-3 curves with involution, and the third one is obtained from a Prymian byMukai's flop. They have the same singularities as two of Fujiki's examples,namely, 28 isolated singular points analytically equivalent to the Veronesecone of degree 8, but a different Euler number. The family of curves used inthis construction forms a linear system on a K3 surface with involution. Thestructure morphism of both Prymians to the base of the family is a Lagrangianfibration in abelian surfaces with polarization of type (1,2). No example ofsuch fibration is known on nonsingular irreducible symplectic varieties.

NEW SYMPLECTIC V-MANIFOLDS OF DIMENSION FOUR VIA THE RELATIVE COMPACTIFIED PRYMIAN