  We are concerned with the following class of equations with exponential
nonlinearities: $$
  \Delta u+h_1e^u-h_2e^{-2u}=0 \qquad \mbox{in } B_1\subset\mathbb{R}^2, $$
which is related to the Tzitz\'eica equation. Here $h_1, h_2$ are two smooth
positive functions. The purpose of the paper is to initiate the analytical
study of the above equation and to give a quite complete picture both for what
concerns the blow-up phenomena and the existence issue.
  In the first part of the paper we provide a quantization of local blow-up
masses associated to a blowing-up sequence of solutions. Next we exclude the
presence of blow-up points on the boundary under the Dirichlet boundary
conditions.
  In the second part of the paper we consider the Tzitz\'eica equation on
compact surfaces: we start by proving a sharp Moser-Trudinger inequality
related to this problem. Finally, we give a general existence result.
