  We consider the following parabolic system whose nonlinearity has no gradient
structure: $$\left\{\begin{array}{ll} \partial_t u = \Delta u + e^{pv}, \quad &
\partial_t v = \mu \Delta v + e^{qu}, u(\cdot, 0) = u_0, \quad & v(\cdot, 0) =
v_0, \end{array}\right. \quad p, q, \mu > 0, $$ in the whole space
$\mathbb{R}^N$. We show the existence of a stable blowup solution and obtain a
complete description of its singularity formation. The construction relies on
the reduction of the problem to a finite dimensional one and a topological
argument based on the index theory to conclude. In particular, our analysis
uses neither the maximum principle nor the classical methods based on
energy-type estimates which are not supported in this system. The stability is
a consequence of the existence proof through a geometrical interpretation of
the quantities of blowup parameters whose dimension is equal to the dimension
of the finite dimensional problem.
