= General-order Cauchy-Kovalevskaya theorem
A <real analytic> system solved for the highest <normal derivatives>, with <real analytic> <Cauchy data>, has a unique local <real analytic> solution. For common order $m$, its normal form is $\partial_t^mU=F(t,x,\{\partial_t^j\partial_x^\alpha U:j<m,\ j+|\alpha|\le m\})$. An invertible coefficient/Jacobian matrix for the highest <normal derivatives> is the system's <non-characteristic hypersurface> condition. A <real analytic> coordinate change flattens the initial <hypersurface>, and adjoining <derivatives> through order $m-1$ reduces the problem to a first-order <real analytic> system. Uniqueness here is in the <real analytic> class; it does not assert <Hadamard well-posedness> in smooth or Sobolev <norms>.
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