  It is shown that the eigenproblem of any $2\times 2$ matrix Hamiltonian with
discrete eigenvalues is involved with a supersymmetric quantum mechanics. The
energy dependence of the superalgebra marks the disparity between the deduced
supersymmetry and the standard supersymmetric quantum mechanics. The components
of an eigenspinor are superpartners\textemdash up to a $SU(2)$
transformation\textemdash which allows to derive two reduced eigenproblems
diagonalizing the Hamiltonian in the spin subspace. As a result, each component
carries all information encoded in the eigenspinor. We also discuss the
generalization of the formalism to a system of a single spin-$\frac{p}{2}$
coupled with external fields. The unconventional supersymmetry can be regarded
as an extension of the Fulton-Gouterman transformation, which can be
established for a two-level system coupled with multi oscillators displaying a
mirror symmetry. The transformation is exploited recently to solve Rabi-type
models. Correspondingly, we illustrate how the supersymmetric formalism can
solve spin-boson models with no need to appeal a symmetry of the model.
Furthermore, a pattern of entanglement between the components of an eigenstate
of a many-spin system can be unveiled by exploiting the supersymmetric quantum
mechanics associated with single spins which also recasts the eigenstate as a
matrix product state. Examples of many-spin models are presented and solved by
utilizing the formalism.
