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We predict the generation of noncritically squeezed light through the spontaneous rotational symmetry breaking occurring in a Kerr cavity. The model considers a chi(3) cavity that is pumped by two Gaussian beams of frequencies omega1 and omega2. The cavity configuration is such that two signal modes of equal frequency omegas=(omega1+omega2)/2 are generated, these signal fields being first-order Laguerre-Gauss modes. In this system, a spontaneous breaking of the rotational symmetry occurs as the signal field corresponds to a Hermite-Gauss TEM mode. This symmetry breaking leads to the perfect and noncritical (i.e., nondependent on the parameter values) squeezing of the angular momentum of the output TEM mode, which is another TEM mode spatially orthogonal to that in which bright emission occurs.