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In the framework of London theory we study the unconventional magnetic state in two-component superconductors with a finite density of fractional flux vortices stabilized near the surface. We show that the process of vortex entry into the two-component superconductor consists of several steps, while the external magnetic field increases from zero. At the first stage only vortices in one of the order parameter components penetrate and sit at the equilibrium position near the surface. When the magnetic field is increased further, vortices in the second-order parameter component eventually enter the superconductor. Such a complex partial vortex penetration leads to the modification of a Bean-Livingston barrier and a magnetization curve as compared to conventional single-component superconductors. We discuss the possibility of experimental identification of protonic superconductivity in the projected superconducting …