We study Livsic's problem of finding $\phi$ satisfying $X\phi=\eta$, where $\eta$ is a given function and $X$ is a given Anosov vector field. We show that, if $\phi$ is a continuous solution and $X,\eta$ are analytic, then $\phi$ is analytic. We use the previous result to show that if two low-dimensional Anosov systems are topologically conjugate and the Lyapunov exponents at corresponding periodic points agree, the conjugacy is analytic. Analogous results hold for diffeomorphisms.