We study a system of equal-size circular disks, each with an asymmetrically placed pivot at a fixed distance from the center. The pivots are fixed at the vertices of a regular triangular lattice. The disks can rotate freely about the pivots, with the constraint that no disks can overlap with each other. Our Monte Carlo simulations show that the one-point probability distribution of orientations has multiple cusplike singularities. We determine the exact positions and qualitative behavior of these singularities. In addition to these geometrical singularities, we also find that the system shows order-disorder transitions, with a disordered phase at large lattice spacings, a phase with spontaneously broken orientational lattice symmetry at small lattice spacings, and an intervening Berezinskii-Kosterlitz-Thouless phase in between.