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We study the dynamics of the five-parameter quadratic family of volume-preserving diffeomorphisms of $\mathbb{R}^3$. This family is the unfolded normal form for a bifurcation of a fixed point with a triple-one multiplier and is also the general form of a quadratic three-dimensional map with a quadratic inverse. Much of the nontrivial dynamics of this map occurs when its two fixed points are saddle-foci with intersecting two-dimensional stable and unstable manifolds that bound a spherical “vortex-bubble.” We show that this occurs near a saddle-center-Neimark–Sacker (SCNS) bifurcation that also creates, at least in its normal form, an elliptic invariant circle. We develop a simple algorithm to accurately compute these elliptic invariant circles and their longitudinal and transverse rotation numbers and use it to study their bifurcations, classifying them by the resonances between the rotation numbers. In particular, rational values of the longitudinal rotation number are shown to give rise to a string of pearls that creates multiple copies of the original spherical structure for an iterate of the map.