We study the analog of the Cauchy‐type integral for the Laplace vector fields theory in case of a piece‐wise Liapunov surface of integration and we prove the Sokhotski‐Plemelj theorem for it as well as the necessary and sufficient condition for the possibility to extend a given Ho¨lder function from such a surface up to a Laplace vector field. Formula for the square of the singular Cauchy‐type integral is given. The proofs of all these facts are based on intimate relations between Laplace vector held and some versions of quaternionic analysis. © 2004 American Institute of Physics