In this article we study the bivariate truncated moment problem (TMP) of degree $2k$ on reducible cubic curves. First we show that every such TMP is equivalent after applying an affine linear transformation to one of 8 canonical forms of the curve. The case of the union of three parallel lines was solved in Zalar (Linear Algebra Appl 649:186–239, 2022.
https://doi.org/10.1016/j.laa.2022.05.008), while the degree 6 cases in Yoo (Integral Equ Oper Theory 88:45–63, 2017). Second we characterize in terms of concrete numerical conditions the existence of the solution to the TMP on two of the remaining cases concretely, i.e., a union of a line and a circle $y(ay +x^2 + y^2) = 0, a \in \mathbb{R}$ \ $\{0\}$, and a union of a line and a parabola $y(x$ $– y^2) = 0$. In both cases we also determine the number of atoms in a minimal representing measure.
