Let ${gamma}{equiv}{gamma}^{(2n)}$ denote a sequence of complex numbers ${gamma}{00},{gamma}{01},cdots,{gamma}0, 2n,...,{gamma}{2n},0;with; {gamma}{00};>;0,{gamma}{ji}={{overline}{gamma_{ij}}}$,and let K denote a closed subset of the complex plane C. The truncated K complex moment problem entails finding a positive Borel measure $mu$ such that ${gamma}{ij}={int}{{overline}{z}}^{i}z^{j}d{mu};(0{leq};i+j;{leq};2n)$ and supp ${mu}{subseteq};K$. If n=2, then is called the quartic moment problem. In this paper, we give partial solutions for the singular quartic moment problem with rank M(2)=5 and ${{overline}{Z}}Z{in};<1,Z,{{overline}{Z}},Z^{2},{{overline}{Z}}^2>$.