NOVELTY - Obtaining (M1) polysaccharides from oleaginous biomass, involves: (a) heating a mixture of oleaginous biomass and an aqueous medium to 155-165 degrees C; (b) maintaining the mixture at the temperature for 15-25 minutes; and then (c) recovering polysaccharides released from the oleaginous biomass from the mixture.    USE - For obtaining polysaccharides and bio-oils from oleaginous biomass comprising algae biomass (claimed), where the bio-oil includes bio-fuel.    ADVANTAGE - The methods are improved and superior methods of producing bio-fuel and other high-value products from oleaginous biomass such as algae; provides two-step methods for the production of bio-oil and value-added compounds from algal-biomass. The two-stage systems and methods provide controlled extraction of high-value co-products in the course of manufacturing bio-fuel from algae biomass. The systems and methods involve the sequential use of 1) sub-critical water extraction followed by 2) hydrothermal liquefaction of the extracted biomass. The combination of these processes into a single production scheme leads to the production of high quality bio-oils without compromising the quality of the associated high-value co-products.    DETAILED DESCRIPTION - INDEPENDENT CLAIMS are included for the following:    (1) obtaining (M2) bio-oils from oleaginous biomass involving: (a1) heating a mixture of oleaginous biomass and an aqueous medium to 237-243 degrees C; (b1) maintaining the mixture at the temperature for 15-25 minutes; and (c1) recovering bio-oils produced during the step of maintaining;    (2) obtaining (M3) polysaccharides and bio-oils from oleaginous biomass, involving: i) heating an aqueous mixture comprising the oleaginous biomass to a first temperature of 155-165 degrees C; ii) maintaining the aqueous mixture comprising the oleaginous biomass at the first temperature for a first period of 1-25 minutes; iii) separating a liquid fraction of the aqueous mixture comprising the oleaginous biomass from a solid fraction of the mixture; iv) recovering polysaccharides released from the oleaginous biomass from the liquid fraction; v) heating an aqueous mixture comprising the solid fraction to a second temperature of 237-243 degrees C; vi) maintaining the aqueous mixture comprising the solid fraction at the second temperature for a second period of 15-25 minutes; and ii) recovering from the aqueous mixture comprising the solid fraction, bio-oils produced during the steps of v) heating and vi) maintaining; and    (3) a system for bio-oil production, comprising: a first closed reactor configured to receive oleaginous biomass and heat the oleaginous biomass to the first temperature of 155-165 (preferably 160) degrees C for the first period of 15-25 (preferably 20) minutes; and a second closed reactor configured to receive reacted oleaginous biomass solids from the first closed reactor and heat the reacted oleaginous biomass solids to the second temperature of 257-263 (preferably 260) degrees C for the second period of 15-25 (preferably 20) minutes.    DESCRIPTION OF DRAWING(S) - The figure shows a schematic diagram of the sequential hydrothermal liquefaction method.