  Amorphous packings of non-spherical particles such as ellipsoids and
spherocylinders are known to be hypostatic: the number of mechanical contacts
between particles is smaller than the number of degrees of freedom, thus
violating Maxwell's mechanical stability criterion. In this work, we propose a
general theory of hypostatic amorphous packings and the associated jamming
transition. First, we show that many systems fall into a same universality
class. As an example, we explicitly map ellipsoids into a system of `breathing'
particles. We show by using a marginal stability argument that in both cases
jammed packings are hypostatic, and that the critical exponents related to the
contact number and the vibrational density of states are the same. Furthermore,
we introduce a generalized perceptron model which can be solved analytically by
the replica method. The analytical solution predicts critical exponents in the
same hypostatic jamming universality class. Our analysis further reveals that
the force and gap distributions of hypostatic jamming do not show power-law
behavior, in marked contrast to the isostatic jamming of spherical particles.
Finally, we confirm our theoretical predictions by numerical simulations.
