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This paper investigates the downlink (DL) transmit design for massive multiple-input multiple-output (MIMO) low-earth-orbit (LEO) satellite communication systems, where only the slow-varying statistical channel state information is exploited at the transmitter. The channel model for the DL massive MIMO LEO satellite system is established, in which both the satellite and the user terminals (UTs) are equipped with uniform planar arrays. Observing the rank-one property of the channel matrices, we show that the single-stream precoding for each UT is the optimal choice that maximizes the ergodic sum rate. This favorable result simplifies the complicated design of transmit covariance matrices into that of precoding vectors without any loss of optimality. Then, an efficient algorithm is devised to compute the precoding vectors. Furthermore, we formulate an approximate transmit design based on the upper bound on the ergodic sum rate, for which the optimality of single-stream precoding still holds. We show that, in this case, the design of precoding vectors can be simplified into that of scalar variables, for which an effective algorithm is developed. In addition, a low-complexity learning framework is proposed for optimizing the scalar variables. Simulation results demonstrate that the proposed approaches can achieve significant performance gains over the existing schemes.