Complex Gradient Systems

Let $M$ be a complex manifold of complex dimension $n+k$. We say that the functions $u_1,\mathrm{.}\mathrm{.}\mathrm{.}s,u_k$ and the vector fields ${\xi }_1,\mathrm{.}\mathrm{.}\mathrm{.},{\xi }_k$ on $M$ form a \emph{complex gradient system} if ${\xi }_1,\mathrm{.}\mathrm{.}\mathrm{.},{\xi }_k,J{\xi }_1,\mathrm{.}\mathrm{.}\mathrm{.},J{\xi }_k$ are linearly independent at each point $p\in M$ and generate an integrable distribution of $TM$ of dimension $2k$ and $d{u}_{\alpha }({\xi }_{\beta })=0$, \d^c\u_\alpha(\xi_\beta)=\delta_{\alpha\beta} for $\alpha ,\beta =1,\mathrm{.}\mathrm{.}\mathrm{.},k$. We prove a Cauchy theorem for such complex gradient systems with initial data along a $\text{CR}-$submanifold of type $(\text{CRdim},\text{CRcodim})$. 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 ${\xi }_{\alpha }^c={\xi }_{\alpha }-J{\xi }_{\beta }$, $\alpha =1,\mathrm{.}\mathrm{.}\mathrm{.},k$ are holomorphic and satisfy $[{\xi }_{a}lph{a}^c,\overline{{\xi }_{\beta }^c}]=0$ for each $\alpha ,\beta =1,\mathrm{.}\mathrm{.}\mathrm{.},k$.