I am trying to solve the following quadratic diophantine equation in $\mathbb Z[T]$: $$((T+1)X+TY-1-Z)((T+1)X+TY-1+Z)=24XY$$ One has the following trivial solutions: $(X,Y,Z)=(0,Y,\pm(1-TY))$, $(X,0,\pm(1-(T+1)X))$. Can one describe all the solutions of this equation (at least an algorithm to obtain all of them)?

Thanks in advance for any answer.