  We study a system of equal-sized circular discs 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 discs can rotate freely about the
pivots, with the constraint that no discs can overlap with each other. Our
Monte Carlo simulations show that the one-point probability distribution of
orientations shows multiple cusp-like 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.
