  There are considered vector fields and quaternionic $\alpha$-hyperholomorphic
functions in a domain of $R^2$ which generalize the notion of solenoidal and
irrotational vector fields. There are established sufficient conditions for the
corresponding Cauchy-type integral along a closed Jordan rectifiable curve to
be continuously extended onto the closure of a domain. The
Sokhotski-Plemelj-type formulas are proved as well.
