Actually, I need to deduce something about the nature of a vector field if it's curl is known completely without actually solving the partial differential equations.
Let's say we have a vector field $\mathbf B$ which depends only on the distance $r$ (perpendicular to z-axis) and points in positive $z$ direction. [I apologize for not being able to put it in single word, I meant that field $\mathbf B$ is such that it points in positive z axis but it's magnitude changes only when we move radially outward from the z axis (or origin), it is zero at the origin]. Then, if we have something like this $$ \nabla \times \mathbf A = \mathbf B$$ can we conclude that there exist an $\mathbf A$ such that it will have similar symmetry, that is moving circumferentially in $xy$ plane would not result in the change of magnitude of $\mathbf A$? And can we conclude that $\mathbf A$ will form closed loops because it has a non-zero curl? And if we take the above two information to be true we will get $\mathbf A$ as a circular field in $xy$ plane.
It is said that above mentioned conclusions about $\mathbf A$ are true due to symmetry and it is also said that if $\mathbf A$ would not have those properties then $\mathbf B$ will also not have the properties which is given, because they are related by the equation $$\nabla \times \mathbf A = \mathbf B$$
Let's consider another scenario, suppose that we have a vector field $\mathbf F$ in $xy$ plane which is uniform both in magnitude and direction all over the $xy$ plane and points in positive $x$ direction, something like this
.
And again if we have something like this $$ \nabla \times \mathbf V = \mathbf F$$ so, can we say that there exits a $\mathbf V$ such that it will also point in one particular direction like $\mathbf F$ and will have same uniformity like $\mathbf F$ ?
I have described my problem through the means of two simple scenarios, it should be noted that I'm not looking for their solutions. I want to understand how does a symmetrical curl implies a symmetrical operand?
My main question is: [How] can we deduce something about the symmetry of operand given the symmetry of its curl without actually solving the partial differential equations?