Jump boundary-value problem on a contour with elongate singularities

Let $\Gamma$ be an image of the interval $(0,1)$ under one-to-one continuous mapping $\phi: (0,1)\to \mathbb{C}$. If the difference of closure of $\Gamma$ and the very set $\Gamma$ contains more than one point, then we say that $\Gamma$ is a contour with elongate singularities. We study the jump boundary-value problem for analytical functions on that contours and obtain new solvability criteria for it.