A Partial Integral Equation

Is there any means to solve the "partial" integral equation of $u$: $$int_{l} u(x,y) dmathcal{H}^1=F(a,b,c)$$ (where $l={(x,y)|ax+by+c=0, a^2+b^2=1}$ is a straight line in $Bbb{R}^2$ and $mathcal{H}^1$ is the 1-dimension Hausdorff measure) under the following different conditions?

I.F is continuous on some closed subset of the half cylinder $Sigma={(a,b,c)|a^2+b^2=1,c ge 0}$;

II.F is $C^k$ on some open subset of $Sigma$ seen as a differentiable manifold.

I come up with this problem when I tried to transform between integral equations and differential equations. The transformation is easily obtained in 1-variable situation, but I got stuck in higher dimension.

Attempt: Write $x=bs-ac,y=-as-bc$ ($s$ is the length parameter since $a^2+b^2=1$) and the equation is transformed into ($mathcal{L}^1$ is the 1-dimension Lebesgue measure) $$int_Bbb{R}u(bs-ac,-as-bc)dmathcal{L}^1=F(a,b,c)$$ Assume $u$ to be smooth and absolutely Riemann integrable we can then calculate partial derivates about $a,b,c$ respectively on both sides, however it does not seem to help in grabbing $u$ "out of" the integral symbol.

Simplification: You can assume $F$ is globally defined on $Sigma$ if it helps to construct an simpler procedure.

Further Simplification: If $F$ is independent of $a,b$, can we conclude that $u$ has radial symmetry?

Bonus: Since $F$ may not be defined on the whole cylinder, the domain that $u$ can be determined is also interesting, but this question may go too far…