TECHNOLOGY FOCUS - INORGANIC CHEMISTRY - Preferred Components: The catalyst is selected from oxides and silicates of aluminum, magnesium, silicon, hafnium, yttrium, titanium, and zirconium.    TECHNOLOGY FOCUS - ORGANIC CHEMISTRY - Preferred Method: The method further involves simultaneously flowing a co-solvent through the spent adsorbent material under near-critical or supercritical conditions. The regenerating the adsorbent medium comprises contacting the catalyst with the solvent at 250-450 degrees C. The regenerating the adsorbent medium further involves contacting the catalyst with the solvent at pressure of 10-30 mPa; and contacting the catalyst with the gas solvent and co-solvent at 10-30 mPa. The regenerating the adsorbent medium involves contacting the catalyst with the solvent and the co-solvent at 250-450 degrees C. Preferred Components: The co-solvent is selected from carbon dioxide, sulfur dioxide, nitrous oxide, sulfur hexafluoride, hydrocarbons, ethers, esters, ketones, alkyl carbonates, and halogenated hydrocarbons. The solvent is fluid or gas. The solvent is 1-5C alcohol. Preferred System: The system comprises a device configured and operably coupled to the vessel to recover the solvent and biofuel separately. The system comprises a second pumping device to direct a co-solvent through a preheater and into the vessel under near-critical or supercritical conditions, where the solvent and the co-solvent are capable of reacting with contaminants adsorbed on the spent adsorbent medium; and pressure and temperature control devices configured and operably coupled to the vessel for maintaining pressure and temperature conditions in the vessel such that the solvent and co-solvent are at or above a critical point to convert the contaminants into biofuel and regenerate the adsorbent medium. The vessel comprising the regenerated adsorbent medium is configured for the purification of biofuel from a solution containing biofuel and contaminants. EXAMPLE - No suitable example given.