Integers that satisfy $a^3= b^2 + 4$

Update: This is a Mordell equation and from the ref E_-00004 from this table all the known solutions were provided here :

E_-00004: r = 1   t = 1   #III =  1
          E(Q) = <(2, 2)>
          R =   0.4503206856
           4 integral points
            1. (2, 2) = 1 * (2, 2)
            2. (2, -2) = -(2, 2)
            3. (5, 11) = -2 * (2, 2)
            4. (5, -11) = -(5, 11)

Fine references about this kind of problems are :

• de Jonquières' 1878 paper (french)
• Conrad's paper for simple impossibilities proofs but not only since the theorem $3.3$ is the proof that no other solutions in $\mathbb{Z}$ exists for your equation.

In Jonquières' paper one finds "D'autres fois, mais rarement, on démontre qu'il n'existe qu'une seule solution. C'est ce qui a été fait par Fermat, Euler et Legendre pour les équations $x^3-2=y^2$, $x^3-4=y^2$...¨.

This means that no other solution exist and that this was proved by one or more between Fermat, Euler and Legendre (I'll search references).

$a=5, b=11$ is one satisfying it. I don't think this is the only pair.