Sums of squares over the Fibonacci $\circ$-ring

The paper considers Diophantine equations of the form
$$ X_1^2+[(X_1+1)\tau]^2+\cdots+X_k^2+[(X_k+1)\tau]^2=A, $$
where $X_i,A\in\mathbb Z$ ($A\ge 0$) are rational integers; $k=2,3,4$, $\tau=(-1+\sqrt{5})/2$ is the golden section, and $[*]$ denotes the integral part of a number. For these equations, the solvability conditions are found, and lower bounds for the number of solutions are obtained. The equations considered are closely related to equations of the form
$$ X_1\circ X_1+\cdots+X_k\circ X_k=A, $$
where $\circ$ denotes the Knuth circle multiplication. Bibliography: 18 titles.