  We studied (arxiv: 1001.4679, 1205.1714) properties of spinors in a toy model
in $d=(1+5)$ as a step towards realistic Kaluza-Klein (like) theories in non
compact spaces. ${\cal M}^{(5+1)}$ was assumed to break to an infinite disc
with a zweibein which makes a disc curved on $S^2$ and with a spin connection
field which allows on such a sphere only one massless spinor state. This time
we are taking into account families of spinors interacting with several spin
connection fields, as required for this toy model by the spin-charge-family
theory We are studying possible masslesness of families of spinors: Spinors
regroup into subgroups of an even number of families.
