Let $a,b$ be coprime integers, neither a perfect power, $n,m$ naturals and $x,y$ integers.

Consider the exponential Diophantine equation $$ a^n-b^m=x^3+y^3 \qquad (*) $$

Nontrivial solution satisfies $|x^3| \ne |a^n|,|x^3| \ne |b^m|, |y^3| \ne |a^n|,|y^3| \ne |b^m|$.

Q1 For fixed $a,b$ are there nontrivial solutions with $n$ and $m$ arbitrary large?

Q2 are there infinitely many pairs $a,b$ for which Q1 gives positive answer?

Numerical evidence suggest these are solutions for all $k,b$: $(a,b,n,m)=(3,b,6,12k)$ and $(3,b,12k+6,24)$.

If this is true, likely the reason is polynomial identity or families of polynomial identities.

In case indeed identities give positive answer:

Q3 Are there more than two univariate identities, preferably infinitely many?

Example:

$3^{66}-5^{24} = 3228113450^3+ 31369668984^3$