The system of PDEs in the original post actually defines a connection $D$ on the bundle $\mathcal{E} = TM \oplus \mathfrak{so}(TM)$, by $$ D_X \begin{pmatrix} Y \\ A \end{pmatrix} = \begin{pmatrix} \nabla_X Y + A(Y)\\ \nabla_X A - R(X,Y) \end{pmatrix} $$ for all vector fields $X,Y$ and field $A$ of skewsymmetric endomorphisms of $TM$. Killing vectors are in bijective correspondence with $D$-parallel sections of $\mathcal{E}$, a result due to Kostant and rediscovered by Geroch, who coined the term Killing transport.

Complete integrability is simply the flatness of $D$. The curvature of $D$ is not difficult to calculate and one finds that $D$ is flat if and only if $R(X,Y)Z = \lambda \left( g(Y,Z)X - g(X,Z)Y \right)$, for some $\lambda \in \mathbb{R}$. This is precisely the statement that the metric has constant sectional curvature.

The system of PDEs in the original post actually defines a connection $D$ on the bundle $\mathcal{E} = TM \oplus \mathfrak{so}(TM)$, by $$ D_X \begin{pmatrix} Y \\ A \end{pmatrix} = \begin{pmatrix} \nabla_X Y + A(X)\\ \nabla_X A - R(X,Y) \end{pmatrix} $$ for all vector fields $X,Y$ and field $A$ of skewsymmetric endomorphisms of $TM$. Killing vectors are in bijective correspondence with $D$-parallel sections of $\mathcal{E}$, a result due to Kostant and rediscovered by Geroch, who coined the term Killing transport.

Complete integrability is simply the flatness of $D$. The curvature of $D$ is not difficult to calculate and one finds that $D$ is flat if and only if $R(X,Y)Z = \lambda \left( g(Y,Z)X - g(X,Z)Y \right)$, for some $\lambda \in \mathbb{R}$. This is precisely the statement that the metric has constant sectional curvature.