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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 $ξ_1,...,ξ_k$ on $M$ form a \emph{complex gradient system} if $ξ_1,...,ξ_k,Jξ_1,...,Jξ_k$ are linearly independent at each point $p\in M$ and generate an integrable distribution of $TM$ of dimension $2k$ and $du_α(ξ_β)=0$, $\d^c\u_α(ξ_β)=δ_{αβ}$ for $α,β=1,...,k$. 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 $ξ_α^c=ξ_α-Jξ_β$, $α=1,...,k$ are holomorphic and satisfy $[ξ_alpha^c,\bar{ξ_β^c}]=0$ for each $α,β=1,...,k$.