Equilibria in non regular economies We know that in regular economies general equilibrium theory predicts a finite and odd number of equilibria, using the properties of the excess demand function and the index theorem. 

How about the non-regular economies? From my understanding, these are the economies in which at least one price vector equilibirum generates a singular matrix of price effects. Geometrically, this can be interpreted as the excess demand function having a zero slope at one of the equilibria as shown in the picture below 

Can we say something more about the equilibria, in case they are a finite number? Do they need to be even, odd or there is no restriction?
@Afreelunch gave you the right intuition. Non-regular economies will exhibit an infinite amount of equilibrium points. Intuitively since there is no local uniqueness, the continuum can be expressed as an interval on the real line so there is an infinite number of such points.