Christopher Hoskin suggested to look at the equation
$$\sum_{i \neq j=1}^4 p_i^2 p_j^2 + a \cdot p_1p_2p_3p_4 = b.$$
where $p_i$ is the distance of a point to vertex $i \in \{1,2,3,4\}$ of a regular tetrahedron.When $a =1$ and $b=0$, there are no solutions, as the left hand side is always positive.
When $a=0$, the above equation is just the equation of a sphere in tetrahedral coordinates.
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