2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/23894Half integer Josephson vortices in 0-$π$-junctions, discussed theoretically and observed experimentally, spontaneously appear at the point where the Josephson phase is $π$-discontinuous. The creation of \emph{arbitrary} discontinuities of the Josephson phase has been demonstrated recently. Here we study fractional vortices formed at an arbitrary $κ$-discontinuity, discuss their stability and possible ground states. The two stable states are not mirror symmetric. Furthermore, the possible ground states formed at two $κ$-discontinuities separated by a distance $a$ are investigated, and the energy and the regions of stability of each ground state are calculated. We also show that the ground states may strongly depend on the distance $a$ between the discontinuities. There is a crossover distance $a_c$ such that for $a<a_c$ and for $a>a_c$ the ground states may be qualitatively different.7 figures, submitted to PRB In v.2 one figure is added, and refs are updated In v.3 major revision, many issues fixedSuperconductivitytext