  We demonstrate the generic existence of Dirac cones in the full
Bistritzer--MacDonald Hamiltonian for twisted bilayer graphene. Its
complementary set, when Dirac cones are absent, is the set of magic angles. We
show the stability of magic angles obtained in the chiral limit by
demonstrating that the perfectly flat bands transform into quadratic band
crossings when perturbing away from the chiral limit. Moreover, using the
invariance of Euler number, we show that at magic angles there are more band
crossings beyond these quadratic band crossings. This is the first result
showing the existence of magic angles for the full Bistritzer--MacDonald
Hamiltonian and solves Open Problem No.2 proposed in the recent survey
arXiv:2310.20642.
