Analytic loops with identities of hypospecial type

Let langle Q, cdot, backslash,slash, e rangle be a local analytic loop with the left alternative property (1) x(xy)= (xx)y and the hypospecial property b(p,q)cdot l_{p, q}(xy)= ((b(p,q)cdot l_{p, q}x)slash b(p,q))cdot (b(p, q)cdot l_{p, q}y) tag 2 (where b(p, q)colon Qtimes Qto Q is an analytic function, b(p, e)=b(e, q)=e; l_{p, q}=L^{-1}_{pq}circ L_pcirc L_q, L_xy overset {rm def}to =xy). In a loop langle Q, cdot, backslash,slash, e ranglebreak one can define the scalar multiplication (t_e)_{tin R}colon Qto Q. par Let us substitute the word w(p, q) (written in terms of operations (cdot), (backslash), (slash), (t_e)_R and (e)) for the function b(p, q) in (2), to obtain the identity (2'). The author shows in this paper that the loop langle Q, cdot, backslash, slash, e rangle with the system of identities (1), (2') can be reduced to a special loop (i.e. a loop where all l_{x, y} are automorphisms) or a left Bol loop with additional identities. More explicitly, in the loop langle Q, cdot, backslash,slash, erangle with (1), (2ʹ) one of the following cases holds: Case 1: l_{p, q}(xy)= l_{p, q}(x)cdot l_{p, q}(y) (special property); w(p,q)cdot (xcdot (w(p,q)backslash y))=((w(p,q)cdot x)slash w(p, q))cdot y. Case 2: x(y(xcdot z))=(xcdot(ycdot x))cdot z (left Bol identity); (v(p, q)x)y=v(p, q)cdot (xy) where v(p, q)= (w(p, q))^2cdot (pq)((p^{-1}q^{-1}))^{-1}.