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We studied properties of spinors in a toy model in $d=(5+1)$, when ${\calM}^{(5+1)}$ breaks to an infinite disc with a zweibein which makes a disccurved on an almost $S^2$ and with a spin connection field which allows on sucha sphere only one massless spinor state, as a step towards realisticKaluza-Klein theories in non compact spaces. Previously we allowed on $S^2$ twokinds of the spin connection fields, those which are gauge fields of spins inand those which are the gauge fields of the family quantum numbers, both asrequired for this toy model by the spin-charge-family theory. This time westudy, by taking into account families of spinors interacting with several spinconnection fields, properties of massless and massive solutions of equations ofmotion, with the discrete symmetries ($\mathbb{C}_{{ \cal N}}$, ${\calP}_{{\cal N}}$, ${\cal T}_{{ \cal N}}$) included. We also allow nonzero vacuumexpectation values of the spin connection fields and study the masses.

Properties of families of spinors in $d=(5+1)$ with zweibein of an  almost $S^2$ and two kinds of spin connection fields, allowing massless and  massive solutions in $d=(3+1)$