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Pairs B, of divergence-free vector fields with compact support in ℛ3 are considered higher-order analog M(B, ) (of order 3) of the Gauss helicity number H(B, ) = ∫A, dℛ3, curl(A) = B; (of order 1) is constructed, which is invariant under volume-preserving diffeomorphisms. An integral expression for M is given. A degree-four polynomial m(B(x1), B(x2), (1), (2)), x1, x2, 1, 2 ∈ ℛ3, is defined, which is symmetric in the first and second pairs of variables separately. M is the average value of m over arbitrary configurations of points. Several conjectures clarifying the geometric meaning of the invariant and relating it to invariants of knots and links are stated. Bibliography: 11 titles.