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The aim of this paper is the study of the center-focus and cyclicity problems inside the class X of 3-dimensional vector fields that admit a first integral that leaves invariant any sphere centered at the origin. We classify the centers of linear, quadratic homogeneous and a family of quadratic vector fields F ⊂ X , restricted to one of these spheres. Moreover, we show the existence of at least 4 limit cycles in family F . [ABSTRACT FROM AUTHOR]