The authors develop a constructive method to search for nonalgebraic invariant curves \(f(x,y)=0\) for planar polynomial differential equations \(\dot x=P(x,y),\, \dot y=Q(x,y).\) More concretely, they search for functions \(f\) of the form \(f(x,y)=\sum_{k=\ell}^n f_k(x,y)=0\) which satisfy \[ P(x,y)\partial f(x,y)/\partial x +Q(x,y)\partial f(x,y)/\partial y =K(x,y)f(x,y), \] where each \(f_k\) is a homogeneous function of degree \(k\) and \(K(x,y)\) is some polynomial of degree at most \(m-1,\) where \(m\) is the maximum of the degrees of \(P\) and \(Q.\) Note that the above set of invariant curves contains the algebraic invariant curves, but as the authors show it contains also other functions. They study separately the case \(xQ_m(x,y)-yP_m(x,y)\equiv0\) (the equator of the Poincaré compactification is full of critical points) and the case \(xQ_m(x,y)-yP_m(x,y)\not\equiv0,\) where \(P_m\) and \(Q_m\) denote the homogeneous parts of degree \(m\) of \(P\) and \(Q,\) respectively. They apply their results to study cubic Lotka-Volterra systems, present systems satisfying \(xQ_3(x,y)-yP_3(x,y)\not\equiv0\) with some nonalgebraic invariant curves involving the exponential integral function \(Ei(x)=-\int_{-x}^\infty \exp(-t)/t\,dt.\) The case of Lotka-Volterra systems satisfying \(xQ_m(x,y)-yP_m(x,y)\equiv0\) is treated in the last section of the paper.