all=Tuples[P[a,-a,b,-b],5] //. P[l___,x_,y_,r___] /; x+y==0:> P[l,r]

Select[all, Length[#]==3&] // Length

Graphics@Select[all, Length[#]==3&] /. P[x_,y_,z_] :> Point[x+(y/3)+(z/9)]/. {a->{1,0}, b->{0,1}}