problem (0. 2) was the same u that of problem (0. 1). Incidentally,
later on Mandzhavidze and Khvedclidze (I) and Simonenko (I) achieved a
direct reduction of problem (0. 2) to problem (0. 1) with the help of
conformal mappings. Apparenlly, the first paper in which SIES were
considered was the paper by Vekua (2) published in 1948. Vekua verified
that the equation (0. 3) where (1; C(f), 5 is the operator of 'ingular
integration with a Cauchy kernel (Srp)(!) "" (". i)-I fr(T -
t)-lrp(T)dT, W is the shift operator (WrpHt) = rp{a(t, in the case 01
= - (13,0, = 0., could be reduced to problem (0. 2). We note thai, in
problem (0. 2), the shift ott) need not be a Carlemao shift, . ei., it
is oot necessary that a . . (t):::: t for some integer 11 2, where ai(l)
"" o(ok_dt)), 0(1(1):::: !. For the first time, the condition 0, (1) ==
1 appeared in BPAFS theory in connection with the study of the problem
(0. 4) by Carle man (2) who, in particular, showed that problem (0. 4)
Wall a natural generalization of the problem on the existence of an a.
utomorphic function belonging to a certain group of Fucs. Thus, the
paper by Vckua (2) is also the fint paper in which a singular integral
equation with a non-Carieman 5hifl is on c sidered.