Skrypnyk, T.Manojlović, Nenad2021-06-222021-06-222021-06http://hdl.handle.net/10400.1/16335In the present paper we develop the algebraic Bethe ansatz approach to the case of non-skew-symmetric gl(2) circle times gl(2)-valued Cartan-non-invariant classical r-matrices with spectral parameters. We consider the two families of these r-matrices, namely, the two non-standard rational r-matrices twisted with the help of second order automorphisms and realize the algebraic Bethe ansatz method for them. We study physically important examples of the Gaudin-type and BCS-type systems associated with these r-matrices and obtain explicitly the Bethe vectors and the spectrum for the corresponding quantum hamiltonians in terms of solutions of Bethe equations. (C) 2021 The Author(s). Published by Elsevier B.V.engYang-baxter equationTriangular boudariesIntegrabilityChainBCSjournal article10.1016/j.nuclphysb.2021.1154241873-1562