Are there vector field solutions of Curl(F) = F in toroidal or bispherical coordinates which are everywhere finite valued and converge to zero at infinity?
More exactly, consider the three functions which satisfy the set of three simultaneous linear first order PDE's implied by the expression curl(F) = F with no finite boundary conditions. These functions, which give the components of F, should be expressed in terms of known functions to provide some intuitive understanding of the space of solutions and provide a basis for proving some theorems about that space.
Solutions of Curl(F) = F without finite boundary conditions exist in at least three coordinate systems: rectangular, cylindrical and spherical. In rectangular coordinates (x,y,z) the solutions are of the form (0, sin(x), cos(x)). In cylindrical coordinates the form of a solution can be obtained by replacing the sin and cos with first and zero order Bessel functions of the radius respectively. In spherical coordinates, the solutions have been expressed in terms of spherical Bessel functions by Chubykalo and Espinoza in "Journal of Physics A Math 35" in 2002. Those solutions are finite valued go to zero in all directions but not fast enough to localize the function to get an square integrable solution. Are there such solutions (finite valued and with no boundary conditions except convergence to zero at infinity and which can be expressed in terms of known functions) in general orthogonal curvilinear coordinate systems?
Special emphasis is on toroidal and bispherical coordinates. Definitions of these coordinate systems can be found online by using those names as search terms, or in "Methods of Theoretical Physics" by Morse and Feshbach and also in "Field Theory Handbook" by Moon and Spencer. As explained by Morse and Feshbach, use of a special coordinate system is a guess that a solution can be found with integral lines that coincide with the coordinate lines thus permitting a simple expression for that solution. I have not been able to find any way to separate the variables in these coordinates for the expression curl(F) = F.
Here is something that may help in finding such solutions. Since F is a curl, it must be that div(F) = 0 and its integral lines are closed curves or can terminate only at a singularity or infinity. In some interesting special cases the above linear system may be reduced to a second order linear ODE for which there is an abundance of theory concerning how the values are related to its singular points. Riemann believed (and proved?) that the singular points of complex valued second order ODEs with analytic coefficients, determine the form of their solutions. Thus the location of the singular points of the coordinate system may play a strong, possibly dominant, role in forming the shape of the solution. For solutions which have simple expressions in the more complicated coordinate systems (as those emphasized above), it is not unreasonable to expect that theorems related to singular points of linear ODEs and Riemann's idea may carry over to the singularities of those coordinates. In the coordinate systems mentioned above, in which known solutions exist, there is at most one singular point or at most one singular line.
In other orthogonal curvilinear systems there are more than one singular point and/or more than one singular line which might help to localize the solution. For example, in bispherical coordinates the two foci are singular points and in an infinitesimal neighborhood around them, the metric is the same as spherical coordinates so there are solutions like those of spherical coordinates in the infinitesimal focal neighborhoods. These solutions can be continued (but numeric solutions are of little value in carrying out the objecrtives mentioned above.) The generally spherically shape surfaces of fixed value of a larger size than those which surround a focal point in bispherial coordinates may form a region containing most of the value when they crunch into the corresponding surfaces around the other focal point. Thus a more localized solution may be created.
Questions:
Is this reasoning correct or are there flaws?
Are there solutions F (in terms of known functions) of the expression curl(F) = F for toroidal or bispherical coordinates?
Are there finite valued solutions F (in terms of known functions and filling all space) of the expression curl(F) = F for toroidal or bispherical coordinates and which converge to zero at infinity? If so, what are they?
Are there reasons for thinking there are no such solutions?