Summary: Under investigation in this paper is a \((2+1)\)-dimensional generalized combined Calogero-Bogoyavlenskii-Schiff-type equation for certain nonlinear phenomena in biohydrodynamics, fluids and plasmas. A novel differential form Lax pair with certain conditions is constructed with an arbitrary function \(H (y, t)\), while an iterative operator different from those in the existing literatures is derived and fully verified to satisfy the above Lax pair. Based on the iterative operator, \(n\)-fold DT and gDT are derived to obtain the higher-order solutions with \(n\) being a positive integer. The \(n\)th-order solutions are influenced by the form of \(H (y, t)\) in the Lax pair, such as the dark solitons with \(H (y, t)\) being a log function whereas the bright solitons with \(H (y, t)\) being a completely quadratic function. Additionally, the solution’s format may be impacted by the alteration in the Lax pair’s form caused by \(H (y, t) = 0\) under specific circumstances. Two types of the many conservation laws are generated separately by discovering two identity relations, which are independent of the form of \(H (y, t)\). It is beneficial in providing a range of physical information for subsequent studies, such as modeling of neural network models for certain fluids as described in this equation.