  We consider an analogue of Kontsevich's matrix Airy function where the cubic
potential $\mathrm{Tr}(\Phi^3)$ is replaced by a quartic term
$\mathrm{Tr}(\Phi^4)$. Cumulants of the resulting measure are known to
decompose into cycle types for which a recursive system of equations can be
established. We develop a new, purely algebraic geometrical solution strategy
for the two initial equations of the recursion, based on properties of Cauchy
matrices. These structures led in subsequent work to the discovery that the
quartic analogue of the Kontsevich model obeys blobbed topological recursion.
