By definition, a Poincare section for the flow associated to a system of n first order differential equations is a (n-1)-dimensional surface (usually a hyperplane in R^n), that is transversal to the flow.
To choose a right Poincare section, not just a hyperplane of equation x_i =0 (or y_i=0 or zi_0) you should associate the 6d vector field,
V(x_1, x_2, y_1, y_2, z_1, z2), having as coordinates the right-hand sides of your system.
The general equation of a hyperplane in R^6 is Ax_1+B_x_2+Cy_1+Dy_2+Ez_1+Fz_2+G=0, and its normal is the vector, N, of coordinates [A, B, C, D, E, F]. If the dot product
of the vector field V and the normal, N, is nonzero
at any point of the hyperplane, <V(x_1, x_2, y_1, y_2, z_1, z_2), N> \neq 0, then that hyperplane is a right choice as a Poincare section.
Hence assigning particular values to A,B, C, D, E, F, you should find a Poincare section, if any.