Polynomial maps with nilpotent Jacobians in dimension three I

In this paper, we first prove that $u,v,h$ are linearly dependent over ${\bf K}$if$JH$isnilpotentand$H$hastheform:$H=(u(x,y,z),v(u,h),h(x,y))$with$H(0)=0$or$H=(u(x,y),v(u,h),h(x,y,z))$with$H(0)=0$. Then we classify polynomial maps of the form $H=(u(x,y),v(x,y,z),h(x,y))$ in the case that $JH$ is nilpotent and $({\mathrm{deg}}_{y}u,{\mathrm{deg}}_{y}h)\le 2$.