I am looking for a coordinate system for the sphere that has constant Lamé parameters. In fact, the Lamé coefficients of the usual spherical coordinate system are:
$L_1 = R$
$L_2 = R\sin(\phi)$
As you can see, $L_2$ is linked to $\phi$.
here is the definition of Lame' coefficients: http://en.wikipedia.org/wiki/Curvilinear_coordinates
Sometimes they are named $A$ and $B$. If we define the surface as: $$ \mathbf{r} = \mathbf{r}(\alpha,\beta) $$ where $\alpha$ and $\beta$ are the curvilinear coordinates we have:
$$ L_1 = \|\mathbf{r},_\alpha \|\\ L_2 = \|\mathbf{r},_\beta \| $$
where $\|\square\|$ is the norm
IF you mean the unit sphere in 3-space, then horizontal radial projection from an enclosing cylinder might be what you're looking for:
$$ (\cos t, \sin t, z) \mapsto (\sqrt{1-z^2} \cos t, \sqrt{1-z^2} \sin t, z) $$
(The map from $(t, z)$ to the cylinder is pretty clear).
I'm guessing, though, that you'll be offended by the "stretch" on longitude lines, which may be what you mean by "Lame parameters".
In general, if you want the measure of distance in the domain and codomain in all directions to be the same, then you're asking for an isometry of a region of the plane with a region of the sphere. Gauss's Theorem Egregium shows this is impossible, for the curvature can be computed solely from the metric, and plane has zero curvature while the sphere does not.
If all you want is for the stretch in the coordinate directions to be nice, then you could rephrase your question as "can I clothe a sphere using non-stretch cloth that allows for arbitrary bias-sheer?" The answer then turns out to locally be "Yes"; finding such a mapping involves, I believe, the sine-Gordon equation, but I could be forgetting something. An undergrad at Brown a couple of years ago did an honors project on this topic with someone in the Applied Math department, and his final presentation included showing how to take a strip of cloth he'd cut on the bias, wrap it around a large ball, and snug it up for a perfect fit in the equatorial regions.