Is there any solution to this system of equations where $x,y,z,s,w,t\in\mathbb{Z}$, none are $0$. \begin{align*} x^2+y^2=z^2\\\ s^2+z^2=w^2\\\ x^2+t^2=w^2 \end{align*}
EDIT: Thank you zwim for the answer. Maybe I should explain where this came from. Well these six numbers correspond to another four numbers $a,b,c,d\in\mathbb{Z}$ in such a way that if you take any two numbers among $a,b,c,d$ their difference produces a perfect square. There is six pairs so it corresponds to finding six perfect squares that satisfy the system of equations above. In the end we have a set of numbers that all differ between each other by a perfect square. I now wonder if there is a set of five numbers with such a property.