  We use the macroscopic fluctuation theory (MFT) to evaluate the probability
distribution P of extreme values of integrated current J at a specified time
t=T in the symmetric simple exclusion process (SSEP) on an infinite line. As
shown recently [Phys. Rev. E 89, 010101(R) (2014)], the SSEP belongs to the
elliptic universality class. Here, for very large currents, the diffusion terms
of the MFT equations can be neglected compared with the terms coming from the
shot noise. Using the hodograph transformation and an additional change of
variables, we reduce the "inviscid" MFT equations to Laplace's equation in an
extended space. This opens the way to an exact solution. Here we solve the
extreme-current problem for a flat deterministic initial density profile with
an arbitrary density 0<n<1. The solution yields the most probable density
history of the system conditional on the extreme current, and leads to a
super-Gaussian extreme-current statistics, - ln P = F(n) J^3/T, in agreement
with Derrida and Gerschenfeld [J. Stat. Phys. 137, 978 (2009)]. We calculate
the function F(n) analytically. It is symmetric with respect to the
half-filling density n=1/2, diverges as n approached 0 or 1, and exhibits a
singularity F(n) |n-1/2| at the half-filling density n=1/2.
