Intersection of Plane x=y and Surface in R3

[autre]

What does a surface in R3 that intersects plane x = y at a line for every value of x represent?

My first intuition is that it represents a plane because in R3 planes intersect at lines but I feel like there is a counterexample to this.

 

[SammyS]

It doesn't need to be s plane.

The surface z = x² intersects the plane y = x for every value of x.

So does x = z³.