The following question was asked at MSE without any solution:

Show that the equation $x^3+y^3+z^3-2xyz=1$ have infinitely many integer solutions $(x,y,z)$.

A more general question was also proposed at MSE:

For which positive integers $N$ does the equation $x^3+y^3+z^3-2xyz=N$ have infinitely many integers solutions $(x,y,x)$?

Some references were given, like the solutions of $x^3+y^3+z^3=nxyz$.

Usually cubic diophantine equations are very hard to deal with.

Does anyone know how any of these two problems could be solved?

Is the second problem (on the equation $x^3+y^3+z^3-2xyz=N$) open?

Any help would be appreciated.