This work is devoted to the study of periodic solutions of generalized Lotka–Volterra systems in three dimensions. The group A3 of isometries of the regular tetrahedron acts on the twelve-dimensional space of such systems. We find four components (modulo the action of A3) of the center variety, i.e. the semi-algebraic subset consisting of systems with a family of periodic solutions (which degenerate at the center). We focus attention on the component which consists of systems X0 with an invariant plane L0 transversal to the coordinate axes and with a (partial) first integral on L0. Under the assumption of normal hyperbolicity of X0 along L0 we begin study of limit cycles appearing for a perturbed system $X_{0}+\varepsilon X_{1}$ (in the Lotka–Volterra class). The linearization of the problem leads to the problem of zeroes of certain Poincaré–Melnikov integrals of a new type. We determine the asymptotic behavior of these integrals near two critical values of the partial first integral of the restricted system X0|L0. We also study the situation near the intersection of different components of the center variety.