  In the paper, we first classify all polynomial maps of the form
$H=(u(x,y,z),v(x,y,z), h(x,y))$ in the case that $JH$ is nilpotent and
$\deg_zv\leq 1$. After that, we generalize the structure of $H$ to
$H=\big(H_1(x_1,x_2,\ldots,x_n),\allowbreak
b_3x_3+\cdots+b_nx_n+H_2^{(0)}(x_2),H_3(x_1,x_2),\ldots,H_n(x_1,x_2)\big)$.
