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The application of a weak integrability concept to the Skyrme and $CP^n$models in 4 dimensions is investigated. A new integrable subsystem of theSkyrme model, allowing also for non-holomorphic solutions, is derived. Thisprocedure can be applied to the massive Skyrme model, as well. Moreover, anexample of a family of chiral Lagrangians providing exact, finite energySkyrme-like solitons with arbitrary value of the topological charge, is given.In the case of $CP^n$ models a tower of integrable subsystems is obtained. Inparticular, in (2+1) dimensions a one-to-one correspondence between thestandard integrable submodel and the BPS sector is proved. Additionally, it isshown that weak integrable submodels allow also for non-BPS solutions.Geometric as well as algebraic interpretations of the integrability conditionsare also given.Comment: 23 page

New integrable sectors in the Skyrme and four-dimensionalCPnmodels

New integrable sectors in the Skyrme and four-dimensionalCPⁿmodels