Let be given functions $f_1,...,f_n:\mathbb{R}^n \to \mathbb{R}$ such that the Jacobian $J=(\partial f_j / \partial x_k)_{j,k=1,\ldots, n}$ exists (say, for all $x\in \mathbb{R}^n)$ and let $a_{ijk} \in \mathbb{R}\, (i=1,..,N,j,k=1,...,n)$ be a sequence of reals. Consider the following system of partial differential equations: $$\tag{GCR} \sum_{j,k=1}^n a_{ijk}\frac{\partial f_j}{\partial x_k}=0\qquad (i=1,...,N)$$ Example: $N=n=2$ and $$\frac{\partial f_1}{\partial x_1}- \frac{\partial f_2}{\partial x_2}=0,\quad\frac{\partial f_1}{\partial x_2}+ \frac{\partial f_2}{\partial x_1}=0\tag{CR}$$ are just the famous Cauchy-Riemann equations. I therefore call (GCR) Generalized Riemann-Cauchy equations. It's well-known that each solution of (CR) is $C^\infty$ (though only existence of first order derivatives is assumed!). This motivates my
Question: Are there known conditions (except CR) on $N$ and $(a_{ijk})$ such that every solution of (GCR) is $C^\infty$ ?
