  While the argument by Zamolodchikov and Polchinski suggests global conformal
invariance implies Virasoro invariance in two-dimensional unitary conformal
field theories with discrete dilatation spectrum, it is not the case in more
general situations without these assumptions. We indeed show that almost all
the globally conformal invariant differential equations in two dimensions are
neither Virasoro invariant nor Weyl invariant. The only exceptions are the
higher spin conservation laws, conformal Killing tensor equations and the
Laplace equation of a conformal scalar.
