Suppose $\sigma$ is a real, symmetric and positive matrix ($\sigma > 0$). The elements of $\sigma$ satisfy a certain number of quadratic equations, this number is less than the number of independent elements in $\sigma$. Moreover, $\sigma$ satisfies an additional constraint:
$$\sigma + i \Omega \ge 0,$$
where $\Omega$ is a real matrix.
Is there a tool which can determine whether $\sigma$ has a solution? And if a solution exists, can we find at least one such solution? This may be more of a programming question, so I don't know if this is the right place to ask.
Additional details which may be useful:
$\sigma$ is an $8 \times 8$ matrix, so it has 36 independent elements.
The number of equations in the elements of $\sigma$ is 16.
$\Omega$ is the symplectic form, i.e. for all symplectic real matrices $S$, $S \Omega S^T = \Omega$.