What is the smallest number $n$ which can be described with equation $(1)$ and one or more of equations $(2.1)-(2.3)$, where all variables are prime numbers? $$a^3 + b^3 + c^3 = n \tag{1}$$ $$d^2 + e^2 -1 = n \quad\tag{2.1}$$ $$f^2 + f \cdot g + g^2 = n \tag{2.2}$$ $$h = n^4 + (n-1)^4 + (n+1)^4\tag{2.3}$$
Is there a second solution for all four equations together? I let the computer check the first million of primes without success.
One solution: $$a=b=7, \quad c=11$$ $$d=13, \quad e=43$$ and, quite fittingly, $$ n=2017. $$