Class of Surfaces of revolution Does exist a kind of surfaces of revolution, in isothermal coordinates (i.e. $E=G$; $F=f=0$, where $E,G,F$ are first fundamental form coefficients, and $e,f,g$ are second fundamental form coefficients) such that $H^2-K=c^2$ (with $H$ nonconstant mean curvature, $K$ nonconstant Gaussian curvature, and $c>0$ constant) ?
Thank you for any help!
 A: With the usual differential geometry notation
$$ H^2 - K = c^2; \quad ( (\kappa_1+\kappa_1)/2)^2 -\kappa_1 \kappa_1 =c^2 \rightarrow (\kappa_1 -\kappa_2)  = \pm  2 c $$
there are two particular cases. First 
$$  c=0 $$
are spheres
EDITS 1-4:
and second non-zero case is a class of surfaces, (shall have to check its name, may be found in book by LP Eisenhart? Blashke? ) which cannot be fully determined if you do not further define its character viz., geodesy, normal curvature /asymptotic conditions.
However your question relates to (principal) curvatures only, so we are talking about their meridians only.You should be looking at meridians only, I am adding meridians a note in  edit to include meridians of DeLaunay and my ( for want of a better name ..Coiloids)
In the pictures below I added arbitrary disposition of fiber angle relations..
as geodesics from a bundle in the first figure and loxodromes in the second figure bundle.You should be looking at meridians only in the figures,the extra rotational symmetry build-up is not directly related to your question. The meridians are isolated in what follows..
The first figure is  a torus among a series of periodic bellows  $\kappa_1 - \kappa_2 = 2 c =1, $ cuspidal minimum radii are  $ =2.4\,  c > c $ 
The second figure writes loxodromes $\psi= 1.5 $ on a meridian with negative 
$\kappa_1 - \kappa_2 =-2.4 c $
Unduloids of DeLaunay and Coiloids 
If $\phi$ is tangent rotation, $u= \cos\phi$,$r$ is radius, $ c\rightarrow 1/c$ ( more comfortable with a physics linear dimension agreement). The ODEs are different and should be given different names. It is quite incorrect to classify or bracket $ y^{\prime\prime}+y =0,\, y^{\prime\prime} - y =0,\, $ together because change of sign in curvature sets them fundamentally apart in Riemannian geometry. Their distinctive ODEs can be written so:
DeLaunay Unduloids
$$ \frac{du}{dr} +\frac{u}{r}=\frac{2}{c}$$
Integrates to
$$ u= \frac{r}{c}+\frac{\lambda}{r} $$
$$ \kappa_1= \frac{1}{c}-\frac{\lambda}{r^2} $$
$$ \kappa_2= \frac{1}{c}+\frac{\lambda}{r^2} $$
Coiloids
$$ \frac{du}{dr}-\frac{u}{r} = \frac{2}{c}$$
Integrates to
$$ u = \frac{2 r\, \log\,r}{c}+ \lambda\,r $$
$$ \kappa_1= \frac{2(1+\log\,r)}{c}+\lambda\, $$
$$ \kappa_2= \frac{2 \log\,r}{c}+\lambda $$

some meridians of the latter class of scalar invariants $ \kappa_1 - \kappa_2= 1/c $

Mohr Circles of curvature
Three curvatures ( compare stress and curvature tensors) is of fundamental importance. Mohr circles of three types ( H,K,D ) respectively. $D$ accounts for failure stress in mechanics of materials and it is (to me ) comforting to treat it as curvature tensor in its own right.
$$ H^2 - D^2 = K $$

If it is a new class to be identified for reference or for referential convenience then the Mohr's circle of curvature of $ \kappa_n $ vs $ \tau_g $ needs to be named ( time being I call it D) and accordingly sketched  to appreciate what remains invariant in this class. $ \kappa_1- \kappa_2 =1/c = D $ is the radius of each of Mohr's circle shown schematically in the third sketch. Mohr's circles of curvature in this $D$ constant case show that any line drawn on rotationally symmetric surface has a common maximum geodesic torsion $\pm  \tau_g = D $ as envelope of sliding Mohr circles.
