When chromatic dispersion operates together with two-dimensional diffraction, the degenerate optical para-metric oscillator exhibits three-dimensional (3D) localized structures in a regime devoid of Turing or modu-lational instabilities. They consist of single lamella, cylinder, or light drops. These 3D structures are generated spontaneously from a weak random noise. We construct 3D bifurcation diagrams associated with these struc-tures. We show that a single cylinder and a single light drop exhibit an overlapping domain of stability. Finally, for a large input intensity, we identify a self-pulsing behavior affecting the stability of 3D localized structures
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