We classify the curves on $S^2$ with fixed monodromy operator and nowhere vanishing geodesic curvature. The number of connected components of the space of such curves turns out to be 2 or 3 depending on the corresponding monodromy. This allows us to classify completely symplectic leaves of the Zamolodchikov algebra, the next case after the Virasoro algebra in the natural hierarchy of the Poisson structures on the spaces of linear differential equations.