  We first construct a counterexample of a generic quadratic submanifold of
codimension $5$ in $\Bbb C^9$ which admits a real analytic infinitesimal CR
automorphism with homogeneous polynomial coefficients of degree $4.$ This
example also resolves a question in the Tanaka prolongation theory that was
open for more than 50 years.
  Then we give sufficient conditions to generate more counterexamples to the
$2-$jet determination Chern-Moser Theorem in higher codimension. In particular,
we construct examples of generic quadratic submanifolds with jet determination
of arbitrarily high order.
