A constructive solution shows that for projectively smooth Weierstrass cubics with one real point at infinity, a representing measure with rank(M) atoms exists whenever any representing measure exists, plus an example needing rank(M(3))+1 atoms and a symmetric-case solution.
Yoo , A.\ Zalar : Constructive approach to the truncated moment problem on reducible cubic curves: Hyperbolic type relations, arXiv preprint https://arxiv.org/pdf/2510.15131
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A constructive approach to the truncated moment problem on cubic curves in Weierstrass form
A constructive solution shows that for projectively smooth Weierstrass cubics with one real point at infinity, a representing measure with rank(M) atoms exists whenever any representing measure exists, plus an example needing rank(M(3))+1 atoms and a symmetric-case solution.