  For an integer $k\geq 2$, let $(F_{n}^{(k)})_{n}$ be the $k-$Fibonacci
sequence which starts with $0,\ldots,0,1$ ($k$ terms) and each term afterwards
is the sum of the $k$ preceding terms. In this paper, we search for powers of 2
which are sums of two $k-$Fibonacci numbers. The main tools used in this work
are lower bounds for linear forms in logarithms and a version of the
Baker--Davenport reduction method in diophantine approximation. This paper
continues and extends the previous work of \cite{BL2} and \cite{BL13}.
