Vector Fields that are Gradients

Note the Lines of Symmetry

There are four vector fields at play. Can you guess what their equations are?

There are only two vector fields plotted above. Can you guess their equations in 2D?

What about these?
http://kevinmehall.net/p/equationexplorer/vectorfield.html#xi+xj%7Cxi+yj%7Cyi+yj%7Cyi+xj%7C%5B-10,10,-10,10%5D