Constants of partial derivations and primitive operations

We describe algebras of constants of the set of all partial derivations in free algebras of unitarily closed varieties over a field of characteristic 0. These constants are also called eigenpolynomials.
It is proved that a subalgebra of eigenpolynomials coincides with the subalgebra generated by values of commutators and Umirbaev–Shestakov primitive elements $p_{m,n}$ on a set of generators for a free algebra.
The space of primitive elements is a linear algebraic system over a signature $\Sigma=\{[x,y],p_{m,n}\mid m,n\ge1\}$. We point out bases of operations of the set $\Sigma$ in the classes of all algebras, all commutative algebras, right alternative and Jordan algebras.