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Strolling struggling on:

This is an inhomogeneous equation. Following the theory we try to
find the general solution in an implicit form:

It is proven that the solution found in this form is indeed
general, i.e. we are not losing any solutions on the way.

Now we can write the associated system (in the symmetrical form):

Or more conveniently for this example, in the canonical form:

Multiplying these equations by l, m and n respectively and summing we get:

This is one of the first integrals of the system. Now multiplying the equations by x, y and z respectively amd summing we obtain:

Therefore, the general solution has the following form:

Geometrically the first solution represents a plane in a 3d space with angular coefficients of the normal vector . The second integral represents a sphere centered at the origin. Therefore, the characteristics of the equation (the curves,

resulting from intersection of these surfaces) are the circles centered on the line passing through the origin with the above mentioned angular coefficients.

Indeed, another way to look at it is rewrite the equation in the following form:

$$\left|\begin{array}{ccc}

\frac{\partial u}{\partial x} & \frac{\partial u}{\partial y} & \frac{\partial u}{\partial z}\\

l & m & n\\

x & y & z\end{array}\right|=0$$

Or:

Where is a tangential vector to the surface , is the vector of the axis of revolution and is the radius vector of an arbitrary point on the surface. It means that for every point on the surface the tangent vector must lie in the plane

passing through the axis of revolution. This is natural, for the surface is ontained by rotating a plane curve against the axis.