Consider the SVD flow on \(\text{GL}(n,\mathbb{R})\) given by: \[ \dot g = g(\pi_0(F(g^Tg)))-(\pi_0(F(gg^T)))g, \] where \(F\) is a smooth real-valued function on \((0,\infty)\) and \(\pi_0\) is the operator on \(n\times n\) matrices such that \(\pi_0(M)=M_- -M_-^T\) with \(M_-\) denoting the strict lower triangular part of \(M\). These flows preserve the singular values of a matrix. Further, if we consider the SVD algorithm as a map, then this map is just the time-one map of the flow corresponding to the choice \(F(x)=\log x\) in the above equation. In this paper, the author establishes the complete integrability of SVD flows for initial data lying in an open, dense subset of \(\text{GL}(n,\mathbb{R})\). The required number of Poisson commuting integrals is given by certain eigenvalues of the following generalized eigenvalue problem: \[ (g)_kw_{rk}=\lambda_{rk}((g^T)^{-1})_k w_{rk}. \] Here \((M)_k\) denotes the \((n-k)\times(n-k)\) matrix obtained from \(M\) by deleting its first \(k\) columns and last \(k\) rows. The variables conjugate to the conserved quantities are obtained from the first components of the generalized eigenvectors \(w_{rk}\), suitably normalized. The level set of integrals is shown to be a product of lines and circles, with the number of circles equal to the number of pairs of complex conjugate eigenvalues \(\lambda_{rk}\).