Alcohol-water mixtures are well known to possess a number of thermodynamic and transport properties that deviate from those of an ideal binary mixture. While the structure of the liquid mixtures is known in detail, the low temperature and high-pressure side of the methastable phase diagram is rather unexplored, even for the simplest amphiphilic molecule, namely methanol, in water. Around the eutectic composition, the liquidus reaches 150 K. Here the high viscosity of the liquid hampers the crystallization, and the mixture shows a tendency to supercool and form a glass. The aim of this proposal is twofold: (i) to determine the structure of the methanol-water system at the eutectic concentration in the low T (80-150 K ) and high- P (1-5 GPa) side of the solid phase diagram, and (ii) to tackle the existence of the expected polyamorphism, as a function of pressure at the same concentration.