New $\mu$-Zariski pairs of surface singularities

To the best of the authors' knowledge, all previously known examples of $\mu$-, $\mu^*$-, link-, or ordinary Zariski pairs of surface singularities in $\mathbb{C}^3$ consist of (possibly weighted) L\^e-Yomdin singularities. In this paper, we present an example of a $\mu$-Zariski pair involving surface singularities that are not of L\^e-Yomdin type.