  We prove that with a $(2+1)$-dimensional Toda type system are associated
algebraic skeletons which are (compatible assemblings) of particle-like Lie
algebras of dyons and triadons type. We obtain trix-coaxial and dyx-coaxial Lie
algebra structures for the system from algebraic skeletons of some particular
choice for compatible associated absolute parallelisms. In particular, by a
first choice of the absolute parallelism, we associate with the
$(2+1)$-dimensional Toda type system a trix-coaxial Lie algebra structure made
of two (compatible) base triadons constituting a $2$-catena. Furthermore, by a
second choice of the absolute parallelism, we associate a dyx-coaxial Lie
algebra structure made of two (compatible) base dyons, as well as particle-like
Lie algebra structures made of single $3$-dyons. Some explicit examples of
applications such as conservation laws related to special solutions, and an
inverse spectral problem are worked out.
