I would like to ask the broad community what is known about the solutions of diophantine equation $$\frac{u}{v} +\frac{v}{w} +\frac{w}{u} =t$$ where $t,v,u,w\in \mathbb{N}.$

I read a book of W. Sierpinski, $\text{ 250 Problems in Elementary Number Theory}$ and there is that this is still an open problem. Is this true? Could anyone give me some references?

I know from MSE that that equation has a solutions of the form $(u,v,w,t) =(k,k,k,3)$ and $(u,v,w,t)=(k,2k,4k,5)$ and also I know that solutions of this equation need not be of the form $(u,v,w) =(a^2 b, b^2 c ,c^2 a).$

I would like to know what is known about solutions of this equation? Any references to this problem will be welcome.