  The nonlinear diffusion equation $u_t = (u^{- 4/3} u_x)_x$ is reduced by the
substitution $u = v^{- 3/4}$ to an equation with quadratic nonlinearities
possessing a polynomial invariant linear subspace of the maximal possible
dimension equal to five. The dynamics of the solutions on this subspace is
described by a fifth-order nonlinear dynamical system (V.A. Galaktionov).
  We found that, on differentiation, this system reduces to a single linear
equation of the second order, which is a special case of the Lam\'e equation,
and that the general solution of this linear equation is expressed in terms of
the Weierstrass $\wp$-function and its derivative. As a result, all exact
solutions $v(x,t)$ on a five-dimensional polynomial invariant subspace, as well
as the corresponding solutions $u(x,t)$ of the original equation, are
constructed explicitly.
  Using invariance condition, two families of non-invariant solutions are
singled out. For one of these families, all types of solutions are considered
in detail. Some of them describe peculiar blow-up regimes, while others fade
out in finite time.
