Continuation of the authors' paper (RZhMat., 1980, 4B820). Convolution operators with semirational symbols (s.s.) are studied. Uniqueness theorems are proved for logarithmic potentials, as well as compatibility theorems for pairs of equations $(K*f)|E=\varphi$, $f|E=\psi$, where $K$ is a kernel with s.s., $E$ is a sufficiently “sparse” subset of the line, $f$ is an “unknown” function. Versions are considered of the “two constants theorem” of Hadamard, relating to uniqueness properties of operators with s.s.