The boundary integral equation (BIE) describes the dynamics of a curved crystallization front separating liquid melt and solid material. We derive a generalized BIE for the thermal-concentration problem taking into account the nonlinear dependence of the crystallization temperature on solute concentration and the kinetics of atomic attachment at the interface. This equation determines the evolution of the interface function and the equation for crystallization driving force - the melt undercooling at the crystal surface. Our calculations carried out for a dendritic vertex in the form of a paraboloid of revolution have shown that the growth rate of the dendritic tip and its diameter substantially depend on the nonlinear effects under study. In particular, the velocity and diameter of the dendrite tip respectively become greater and narrower with increasing deviation of the liquidus equation from the linear relationship. Also, the dendrite tip velocity can be significantly affected by variations in the exponent of atomic kinetics.