Am I a Secondary Taxicab? Background
Ramanujan's number, \$1729\$, is called a taxi-cab number due to the (possibly apocryphal) tale of Hardy boarding a cab to visit Ramanujan in hospital having this number, which seemed bland to him.
It's since known as the most famous of a class of integers known as &quot;taxicab numbers&quot; which are expressible as the sum of two \$n\$th powers (of positive integers) in two (or sometimes \$k\$) different ways.
\$1729\$ is the smallest natural number expressible as the sum of 2 cubes in 2 different ways, making it the first \$3,2\$ taxicab number (\$n,k\$ being general).
Challenge
Given a number, decide whether it is a \$3,2\$ 'secondary taxicab number' - meaning it fulfils the same constraint as \$1729\$ (2 unique sums of cubes), but does not have to be the smallest such integer of the \$3,2\$ class (that being 1729, of course).
Example cases
$$1729 = 10^3 + 9^3 = 12^3 + 1^3 \\
4104 = 15^3 + 9^3 = 16^3 + 2^3 \\
13832 = 2^3 + 24^3 = 18^3 + 20^3$$
As well as \$20683, 32832, 39312...\$
Scoring
This is code-golf, so the shortest answer in each language wins.

Rough Matlab code to find other cases by brute force:
for k = 1729:20000
    C = sum(round(mod(real((k-[1:ceil(k^(1/3))].^3).^(1/3)),1)*10000)/10000==1);
    if C &gt; 1
        D = (mod(C,2)==0)*C/2 + (mod(C,2)==1)*((C+1)/2);
        disp([num2str(k),' has ',num2str(D),' solns'])
    end
end
Jelly, 9 bytes

Credits to Erik the Outgolfer.

Œċ*3S€ċ&gt;1


Try it online!

This is too slow that it won't even work for 1729 online.

Much faster, 12 bytes

Credits to Dennis.

R*3fRŒċS€ċ&gt;1


Try it online!