Local integrability of Mizohata structures

In this work we study the local integrability of strongly pseudoconvex Mizohata structures of rank n > 2 n > 2 (and co-rank 1 1 ). These structures are locally generated in an appropriate coordinate system ( t 1 , … , t n , x ) ({t_1}, \ldots ,{t_n},x) by flat perturbations of Mizohata vector fields M j = ∂ ∂ t j − i t j ∂ ∂ x {M_j} = \frac {\partial } {{\partial {t_j}}} - i{t_j}\frac {\partial } {{\partial x}} , j = 1 , … , n j = 1, \ldots ,n . For this, we first prove the global integrability of small perturbations of the structure generated by ∂ ∂ z ¯ + σ 1 ∂ ∂ z \frac {\partial } {{\partial \bar z}} + {\sigma _1}\frac {\partial } {{\partial z}} , ∂ ∂ θ n − 1 + σ j ∂ ∂ z \frac {\partial } {{\partial {\theta _{n - 1}}}} + {\sigma _j}\frac {\partial } {{\partial z}} , j = 2 , … , n j = 2, \ldots ,n , defined over a manifold C × S {\mathbf {C}} \times S , where S S is simply connected.