Let $R=\mathbb{R}[X_1,\dots,X_n]$, and $\mathfrak{I}_d=\{ \text{ideals for which there is minimal generating system with } d \text{ elements} \}$ be partially ordered by inclusions. Is there an algorithm or criterion that can determine if a given ideal $I$ is maximal in $\mathfrak{I}_2$. If an ideal is not maximal, then the algorithm should provide a list of all ideals in $\mathfrak{I}_2$ that contain $I$.

This question arose from my previous question. For which Pace Nielsen provided very nice examples: $$I_1:=<x_1 x_2, x_1(x_2^4+x_1^2)-x_2^3(x_1 x_2)>=<x_1x_2,x_1^3> \subsetneq I_2:=<x_1x_2,x_2^4+x_1^2>$$

In this case $I_1$ is not maximal but $I_2$ might be. For example, is $gcd(I)=1$ sufficient?