The work of this paper is a part of an ongoing research on understanding singularities of envelopes for differential equations of Clairaut type. Let us explain the main notions and concepts. First, consider the ordinary differential equation \[ F(x,y,p)=0, \tag{1} \] where \(p\) stands for the derivative \(dy/dx\) and \(F\) is defined in a domain of the space \(J^1(\mathbb{R},\mathbb{R})\) that consists of 1-jets of functions \(y(x)\), i.e., the space with coordinates \(x,y,p\) equipped with the contact 1-form \(pdx - dy = 0\). Assume that equation (1) defines a smooth surface in \(J^1(\mathbb{R},\mathbb{R})\), then the contact structure cuts the vector field \[ X = F_p \partial_x + pF_p \partial_y + (F_x+pF_y) \partial_p \] on this surface. Integral curves of the field \(X\) are 1-jet extensions of solutions of equation (1). The canonical projection \(\pi(x,y,p) = (x,y)\) restricted to the surface \(\{F=0\}\) has singular points on the set \(\{F=F_p=0\}\) called the criminant, and the projection of the criminant is the discriminant set of equation (1). Generically, the criminant and the discriminant set are curves, the field \(X\) vanishes at isolated points of the criminant, and the discriminant set is the locus of singularities of solution of (1) (almost all of which are 3:2-cusps). However, there exist a special class of equations (1) called Clairaut type. It is defined by the condition that the function \(F_x+pF_y\) vanishes on the criminant identically. Under light additional conditions, in this case the discriminant set is the envelop of solutions of (1), and consequently, it is a solution as well. The basic examples are \(p^2 = y\) and classical Clairaut’s equation itself: \[ f(p) = xp-y, \ \ f''(p) \not\equiv 0. \tag{2} \] The discriminant set of (2) is the dual Legendrian curve to the graph \(y=f(x)\), it is the envelop of its tangent lines \(xc-y=f(c)\), \(c=const\), which are also solutions of (2). It is regular at points where \(f''(p) \neq 0\) are it is singular if \(f''(p)=0\). For instance, it has 3:2-cusps at points where \(f''(p)=0\), \(f'''(p) \neq 0\). This is the simplest example of singularities of envelopes.
Second, the authors investigate the partial differential equation \[ F(x_1, x_2,y,p_1, p_2)=0, \tag{3} \] where \(p_i\) stands for the derivative \(dy/dx_i\) and \(F\) is defined in a domain of the space \(J^1(\mathbb{R}^2,\mathbb{R})\) that consists of 1-jets of functions \(y(x_1,x_2)\), i.e., the space with coordinates \(x_1, x_2,y,p_1, p_2\) equipped with the contact 1-form \(p_1 dx_1+ p_2 dx_2 - dy = 0\). Similarly to the above, there exists a special class of equations (3) called Clairaut type. The authors show that (under light additional conditions) Clairaut type equations (3) have envelops of solutions (which are solutions as well) and establish the list of typical singularities of their envelops: cuspidal edge, swallowtail, cuspidal butterfly, cuspidal lips/beaks, etc (frontal singularities).
Third, the authors investigate the system of equations \[ F(x_1, x_2,y,p_1, p_2)=0, \ \ G(x_1, x_2,y,p_1, p_2)=0, \tag{4} \] where the functions \(F, G\) are defined in a domain of the space \(J^1(\mathbb{R}^2,\mathbb{R})\), all notations are similar to (3) and the Poisson bracket \([F,G]\) on the manifold \(\{F=G=0\}\) is identically zero. The authors introduce a natural notion of Clairaut type systems (4) and establish the list of typical singularities of their envelops similar to those for (3).