Summary: This paper concerns the description of the complex analytic, i.e., entire or meromorphic, solutions of the homogeneous first-order partial differential equation \((u_{z_1} )^m + (u_{z_2} )^m = u^m (m \geq 1)\) in \(\mathbb C^2 \). Also, the characterization and classification of the left-factors (in the sense of composition of meromorphic functions of several variables in the factorization theory) of the entire solutions of the homogeneous first-order partial differential equation \((u_{z_1} )^m + (u_{z_2} )^m + \dots + (u_{z_n} )^m = u^m (m \geq 1, n \geq 2)\) in \(\mathbb C^n\) is given, which is complemented by several examples to show the accuracy.