🚀 go-pugleaf

I am interested in certain surfaces in R^3 defined by the following
metric g_ij and second fundamental form L_ij (i,j = 1,2) :

   g_ij = E(x,y) (dx^2 + dy^2)

   L_ij = c (dx^2 - dy^2).

Apologies for the abuse of notation.  The components are

   g_11 = g_22 = E(x,y)

   L_11 = -L_22 = c 

   g_12 = g_21 = L_12 = L_21 = 0

where E(x,y) is a function of x and y, and c is a constant.
This is a minimal surface because the mean curvature is zero:

   K_m = L_11 g_22 + g_11 L_22 = 0.

I want to know whether surfaces of this form have been studied before,
and, if so, references to the literature.

--
Michael S. Ody,
The Physics Laboratory,
University of Kent at Canterbury,
CT2 7NZ,  United Kingdom.