I encountered the formula $$x^3+y^3=z^3+1$$ with the condition, that $$x<y<z$$ and wonder, whether it has got a specific name or whether it can be easily transformed into another well-known (family of) formula(s).
Have a look at http://www.mathpages.com/home/kmath071.htm
There you will find $$(1\pm9m^3)^3+(9m^4)^3+(-9m^4\pm3m)^3=1$$ and another similar-but-more-complicated formula, also it says it is known that there are infinitely many such formulas, and it is not known whether every solution is part of such an infinite family.
$$X^3+ Y^3+ Z^3=1$$ is the formula which is known as harder factor and yours is a distorted and conditional form of harder factor
If $X+Y+Z=0$ then $X^3+ Y^3+ Z^3=1$.
In your question $X$ is less than $Y$ and $Y$ is less than $Z$ means the minimum possible difference between $X$ and $Y$, $Y$ and $Z$ is $1$. At the same time the minimum possible difference between $X$ and $Z$ will be $2$.
So there will in all the cases except $X=-2$, $Y=-1$, $Z=3$ where $X+Y+Z$ is not equal to zero then it must be that $X^3+ Y^3+ Z^3$ is not equal to $1$. So $X^3+ Y^3+ Z^3$ must be greater/less than $1$. As it is given that $Z>Y>X$ then $X^3+ Y^3$ must be unequal to $Z^3$. It means $X^3+ Y^3$ may be equal to $Z^3+1$.
In this way $X^3+ Y^3+ Z^3=1$ is related to the question asked by the poster