Integer solutions to $\frac{d^3}{r}+r=a^2$ 
What are the positive integer solutions $(a,d,r)$ to $\frac{d^3}{r}+r=a^2$?

This is a revised version of my deleted question. Alternate forms are $d^3 = r(a^2-r)$ and from the quadratic formula $a^4-4d^3=b^2 \Rightarrow 4d^3 = a^4 - b^2=(a^2+b)(a^2-b)$, with $a,b$ the same parity.
Small solutions $(d, r, a)$: $(2,1,3), (72,36,102), (75,25,130), (92,8,312),(360,81,759)$

This has an interesting connection to Pell equations. Let,
$$(a^2-r)r = d^3\tag1$$
then this has an infinite number of integer solutions given by, 
$$d,r,a = \tfrac{n}{2}y^2;\;\tfrac{1}{4}y^2,\;\tfrac{1}{2}xy,\tag2$$
where,
$$x^2-2n^3y^2 = 1\tag3$$
Since for $n=1$ (and others) the $y$ are all even $y = 0, 2, 12, 70, 408, 2378, 13860,\dots$ as A001542, then $(2)$ are all integers. Its smallest $d,r,a$ are,
$$2,\,1,\,3\\72,\,36,\,102$$
Of course, $(2)$ does not give all solutions, but easily shows there is an infinity of them.
