  Given real numbers $\beta \equiv \beta ^{\left( 4\right) }\colon \beta_{00}$,
$\beta _{10}$, $\beta _{01}$, $\beta _{20}$, $\beta _{11}$, $ \beta _{02}$,
$\beta _{30}$, $\beta _{21}$, $\beta _{12}$, $\beta _{03}$, $\beta _{40}$,
$\beta _{31}$, $\beta _{22}$, $\beta _{13}$, $\beta _{04}$, with $\beta _{00}
>0$, the quartic real moment problem for $\beta $ entails finding conditions
for the existence of a positive Borel measure $\mu $, supported in
$\mathbb{R}^2$, such that $\beta _{ij}=\int s^{i}t^{j}\,d\mu \;\;(0\leq i+j\leq
4) $. Let $\mathcal{M}(2)$ be the 6 x 6 moment matrix for $\beta^{(4)}$, given
by $\mathcal{M}(2)_{\mathbf{i},\mathbf{j}}:=\beta_{\mathbf{i}+\mathbf{j}}$,
where $\mathbf{i},\mathbf{j} \in \mathbb{Z}^2_+$ and
$\left|\mathbf{i}\right|,\left|\mathbf{j}\right|\le 2$. In this note we find
concrete representing measures for $\beta^{(4)}$ when $\mathcal{M}(2)$ is
nonsingular; moreover, we prove that it is possible to ensure that one such
representing measure is 6-atomic.
