Complex Gradient Systems

Let $M$ be a complex manifold of complex dimension $n+k$. We say that the functions $u_1,...s,u_k$ and the vector fields $\xi_1,...,\xi_k$ on $M$ form a \emph{complex gradient system} if $\xi_1,...,\xi_k,J\xi_1,...,J\xi_k$ are linearly independent at each point $p\in M$ and generate an integrable distribution of $TM$ of dimension $2k$ and $du_\alpha(\xi_\beta)=0$, \d^c\u_\alpha(\xi_\beta)=\delta_{\alpha\beta} for $\alpha,\beta=1,...,k$. We prove a Cauchy theorem for such complex gradient systems with initial data along a $\CR-$submanifold of type $(\CRdim,\CRcodim)$. We also give a complete local characterization for the complex gradient systems which are \emph{holomorphic} and \emph{abelian}, which means that the vector fields $\xi_\alpha^c=\xi_\alpha-J\xi_\beta$, $\alpha=1,...,k$ are holomorphic and satisfy $[\xi_alpha^c,\bar{\xi_\beta^c}]=0$ for each $\alpha,\beta=1,...,k$.