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Using the Gorkov equations, we provide an exact solution for aone-dimensional model of superconductivity in the presence of a conicalhelicoidal exchange field. Due to the special type of symmetry of the system,the superconducting transition always occurs into a nonuniform superconductingphase (in contrast with the Fulde-Ferrell-Larkin-Ovchinnikov state, whichappears only at low temperatures). We directly demonstrate that the uniformsuperconducting state in our model carries a current and thus does notcorrespond to the ground state. We study in the framework of the Bogoliubov-deGennes approach the properties of the Josephson junction with a conicalferromagnet as a weak link. In our numerical calculations, we do not use anyapproximations (such as, e.g., a quasiclassical approach), and we show arealization of an anomalous $\phi_{0}$ junction (with a spontaneous phasedifference $\phi_{0}$ in the ground state). The spontaneous phase difference$\phi_{0}$ strongly increases at high values of the exchange field near theborderline with a half-metal, and it exists also in the half-metal regime.

Nonuniform superconductivity and Josephson effect in conical ferromagnet