The continuous demand for content-rich multimedia is pushing for high-speed and secure transmission approaches. In recent years, linear network coding (LNC) has been shown to be a promising technology to improve network throughput, transmission reliability, and information security. In this paper, we study the optimal transmission topology construction and LNC design for a secure multiple-source multicast to deliver the same content with integral link rates, which can be equivalent to the secure multicast problem with a virtual source, i.e., the integer secure virtual-source multicast (ISVM) problem. The objectives of the ISVM problem include the following: 1) satisfy the weakly secure requirements, 2) maximize the secure multicast rate (SMR), and 3) minimize the transmission cost when the SMR is maximized. First, we analyze the necessary and sufficient condition that there exist a transmission topology with integral link rates and a secure LNC that can achieve a given SMR $R$. Then, we model the ISVM problem as an integer linear programming based on the theoretical analysis and design an efficient transmission topology construction algorithm to solve the ISVM problem by utilizing the Lagrangian relaxation and subgradient method. We also analyze the size of finite field required to construct the deterministic LNC for a secure virtual-source multicast and the probability that the virtual-source multicast is weakly secure when using random LNC in the ISVM problem. Finally, we design upper and lower bounds for the ISVM problem and conduct extensive simulations to compare the performance of the proposed algorithms with these two bounds.