Let $N$ be fixed non-negative integer.

Are there infinitely many integer solutions to

$$ a^4+b^4-c^4=N \qquad (1)$$

(1) is a surface, so possible approach is to find genus 0 curve on it with infinitely many integral points.

Partial argument for positive answer is similar equation is possible over larger ring: $a^4+b^4+c^4=18$ over $\mathbb{Z}[i],i^2=-1$.

Let $a= x - y - 1 ,b= 2*y + 2 ,c= -x - y - 1$. Then $a^4+b^4+c^4=18 + q_1(x,y) q_2(x,y)$ where $q_1=x^2 + 3*y^2 + 6*y$. When $q_1=0$ we have solution and this is possible infinitely often over $\mathbb{Z}[i]$.

Is there non-zero integer $N$ such that

$$ a^4+b^4-c^4=N \qquad (1)$$

has infinitely many integer solutions $(a,b,c)$ with $a,b \ne \pm c$?

(1) is a surface, so possible approach is to find genus 0 curve on it with infinitely many integral points.

Partial argument for positive answer is similar equation is possible over larger ring: $a^4+b^4+c^4=18$ over $\mathbb{Z}[i],i^2=-1$.

Let $a= x - y - 1 ,b= 2*y + 2 ,c= -x - y - 1$. Then $a^4+b^4+c^4=18 + q_1(x,y) q_2(x,y)$ where $q_1=x^2 + 3*y^2 + 6*y$. When $q_1=0$ we have solution and this is possible infinitely often over $\mathbb{Z}[i]$.