In this talk, we propose a novel variant three-field Blaszak-Marciniak (BM) lattice equation by introducing trigonometric-type bilinear operators and leveraging the reduction technique from the variant KP-Toda hierarchy. Applying Hirota's bilinear method, we derive general Gram-type determinant solutions. Rational solutions of the variant BM lattice are obtained via B\"acklund transformation and differential operators method. Higher-order rogue waves are formulated in terms of Schur polynomials, whose peak patterns are linked to the root structures of Yablonskii-Vorob'ev polynomials and Umemura polynomials. Furthermore, three distinct types of breather solutions are identified, and numerical three-periodic wave solutions are computed using the Gauss-Newton method.