What is an energy independent basis?

I am reading this article The use of basis splines and non-orthogonal orbitals in R-matrix calculations: Application to Li photoionization

At some point they span a wave function $Psi_{E}$ in terms of energy independent basis

$Psi_{E}=sum_{k} A_{E k} Psi_{k}$

Here for example The Structures of Binary Compounds it is said that LCAO is a energy independent basis.
Here Bound states of a short-range potential
we have

To calculate the energy spectrum, we
write the energy eigenvalue equation (wave equation) $H|psirangle=E|psirangle$ in the energy independent basis as $sum_{m} f_{m} Hleft|phi_{m}rightrangle=E sum_{m} f_{m}left|phi_{m}rightrangle .$

so it seems that energy independent basis is basis whose vector has the same energy.

My question is what is a energy independent basis?