ThisbookisasequeltoSymmetriesandIntegrationMethods(2002),byGeorgeW.BlumanandStephenC.Anco.ItincludesasignificantupdateofthematerialinthelastthreechaptersofSymmet'riesan,dDzjjerentialEqua-tions(1989;reprintedwithcorrections,1996),byGeorgeW.BlumanandSukeyukiKumei.Theemphasisinthepresentbookisonhowtofindsys-tematicallysymmetries(localandnonlocal)andconservationlaws(localandnonlocal)ofagivenPDEsystemandhowtousesystematicallysymmetriesandconservationlawsforrelatedapplications.Inparticular,foragivenPDEsystem,itisshownhowsystematically(1)tofindhigher-orderandnonlocalsymmetriesofthesystem;(2)toconstructbydirectmethodsitsconserva-tionlawsthroughfindingsetsofconservationlawmultipliersandformulastoobtainthefluxesofaconservationlawfromaknownsetofmultipliers;(3)todeterminewhetherithasalinearizationbyaninvertiblemappingandcon-structsuchalinearizationwhenoneexistsfromknowledgeofitssymmetriesandlorconservationlawmultipliers,inthecasewheiithegivenPDEsystemisnonlinear;(4)touseconservationlawstoconstructequivalentnonlocallyrelatedsystems;(5)tousesuchnonlocallyrelatedsystemstoobtainnonlo-calsymmetries,nonlocalconservationlawsandnon-invertiblemappingstolinearsystems;and(6)toconstructspecificsolutionsfromreductionsarisingfromitssymmetriesaswellasfromextensionsofsymmetrymethodstofindsuchreductions.　　Thisbookisaimedatappliedmathematicians;scientistsandengineersinterestedinfindingsolutionsofpartialdifferentialequationsandiswritteninthestyleoftheabove-mentioned1989bookbyBlumanandKumei.Therearenumerousexamplesinvolvingvariouswell-knownphysicalandengineeringPDEsystems.