This paper introduces Lax pairs informed neural networks (LPNNs) to solve integrable systems, focusing on the development of two versions: LPNN-v1 and LPNN-v2. LPNN-v1 transforms the solving of nonlinear integrable systems into linear Lax pairs spectral problems, efficiently solving data-driven localized wave solutions and obtaining spectral parameters and functions. LPNN-v2 incorporates the compatibility condition/zero curvature equation of Lax pairs, enhancing the accuracy of solving high-precision data-driven localized wave solutions and associated spectral problems. The paper demonstrates the effectiveness of LPNNs through numerical experiments on various integrable systems, including the Korteweg-de Vries (KdV) equation, modified KdV equation, nonlinear Schrödinger equation, sine-Gordon equation, Camassa-Holm equation, short pulse equation, Kadomtsev-Petviashvili equation, and high-dimensional KdV equation. The results show that LPNNs significantly improve training efficiency and accuracy compared to conventional methods, making them a promising tool for studying integrable systems.This paper introduces Lax pairs informed neural networks (LPNNs) to solve integrable systems, focusing on the development of two versions: LPNN-v1 and LPNN-v2. LPNN-v1 transforms the solving of nonlinear integrable systems into linear Lax pairs spectral problems, efficiently solving data-driven localized wave solutions and obtaining spectral parameters and functions. LPNN-v2 incorporates the compatibility condition/zero curvature equation of Lax pairs, enhancing the accuracy of solving high-precision data-driven localized wave solutions and associated spectral problems. The paper demonstrates the effectiveness of LPNNs through numerical experiments on various integrable systems, including the Korteweg-de Vries (KdV) equation, modified KdV equation, nonlinear Schrödinger equation, sine-Gordon equation, Camassa-Holm equation, short pulse equation, Kadomtsev-Petviashvili equation, and high-dimensional KdV equation. The results show that LPNNs significantly improve training efficiency and accuracy compared to conventional methods, making them a promising tool for studying integrable systems.