Rational curves on lattice-polarised K3 surfaces

Fix a K3 lattice $\mathrm{\Lambda }$ of rank two and $L\in \mathrm{\Lambda }$ a big and nef divisor that is positive enough. We prove that the generic $\mathrm{\Lambda }$-polarised K3 surface has an integral nodal rational curve in the linear system $\mathrm{\mid }L\mathrm{\mid }$, in particular strengthening previous work of the first named author. The technique is by degeneration, and also works for many lattices of higher rank.